Debye-Hiickel additivity

For more concentrated solutions (/° 5 >0.3) an additional term BI is added to the equation B is an empirical constant. For a more detailed treatment of the Debye-Hiickel theory a textbook of physical chemistry should be consulted.1... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

Among other applications of electrolyte solution theory to defect problems should be mentioned the application of the Debye-Hiickel activity coefficients by Harvey32 to impurity ionization problems in elemental semiconductors. Recent reviews by Anderson7 and by Lawson45 emphasizing the importance of Debye-Hiickel effects in oxide semiconductors and in doped silver halides, respectively, and the book by Kroger41 contain accounts of other applications to defect problems. However, additional quantum-mechanical problems arise in the treatment of semiconductor systems and we shall not mention them further, although the studies described below are relevant to them in certain aspects. 

In addition, several other forms of correlating equations give comparable fits to the experimental data. One equation uses the higher order limiting law, followed by an empirical polynomial in the square-root of molality. Similarly, another equation uses the Debye-Hiickel limiting law with B set equal to zero, followed by an empirical polynomial in the square-root of molality. Both of these have been discussed in detail elsewhere (Staples and Nuttalf, 1977). 

Using the Debye-Hiickel-Onsager law, Kilpatrick and Luborsky in addition calculated the theoretical curve which, however, is obeyed by the measured values only at the lowest concentrations. These were experimentally accessible in the case of hexamethylbenzene. The values for e(HF) and tjhf for the calculation of the constants were taken from the literature (Fredenhagen and Dahmlos, 1928 Simons and Dresdner, 1944) (62o°c = 59 j2o°c = 0-00210 poise). One thus finds the constant A at 20°C of the Debye-Hiickel-Onsager law to be given by the equation ... 

Assuming again that y follows the Debye-Hiickel law, the total pressure P is measured as a function of the solute concentration, then the vapor phase y, the only unknown in Equation 4, can be calculated, and hence the activities a and a2 can also be calculated, provided the activities ai° and a of each solvent prior to the addition of the solute are known dG°/dZ can be obtained next from Equation 1. Finally, integration of dG°/<9Zi with respect to Z leads to the standard molar free energy of transfer AG°t between Z = 1 (if water is chosen as the reference solvent) and any value of Z. ... 

The Gibbs phase rule is the basis for organizing the models. In general, the number of independent variables (degrees of freedom) is equal to the number of variables minus the number of independent relationships. For each unique phase equilibria, we may write one independent relationship. In addition to this (with no other special stipulations), we may write one additional independent relationship to maintain electroneutrality. Table I summarizes the chemical constituents considered as variables in this study and by means of chemical reactions depicts independent relationships. (Throughout the paper, activity coefficients are calculated by the Debye-Hiickel relationship). Since there are no data available on pressure dependence, pressure is considered a constant at 1 atm. Sulfate and chloride are not considered variables because little specific data concerning their equilibria are available. Sulfate may be involved in a redox reaction with iron sulfides (e.g., hydrotroilite), and/or it may be in equilibrium with barite (BaS04) or some solid solution combinations. Chloride may reach no simple chemical equilibrium with respect to a phase. Therefore, these two ions are considered only to the... 

The forth issue is the increase in the repulsion between bilayers at short distances. In Fig. 1, the osmotic pressure is plotted as a function of separation distance (data from Ref. [13]) for no added salt, for l M KC1 and for 1 M KBr. They reveal an increase in repulsion at short separation distances upon addition of salt. While the relatively small difference between 1 M KC1 and 1 M KBr can be attributed to the charging of the neutral lipid bilayers by the binding of Br (but not C.1-) [14], the relatively large difference between no salt and 1 M KCl is more difficult to explain. Even a zero value for the Hamaker constant (continuous line (2) in Fig. 1), in the 1 M KCl case, is not enough to explain the increase in repulsion, determined experimentally. The screening of the van der Waals interaction, at distances of the order of three Debye-Hiickel lengths (about 10 A) should lead, according to Petrache et al. calculations, to a decrease of only about 30% of the Hamaker constant (from 1.2kT to about 0.8kT, see Fig. 5C of Ref. [14]). Therefore, an additional mechanism to increase the hydration repulsion or the undulation force (or both) upon addition of salt should exist to explain the experiments. 

For hydrochloric acid, a strong electrolyte, calculate an experimental value of with Eq. (9) for each of the concentrations studied. In addition, use Eq. (13) to obtain a value of the osmotic coefficient based on the Debye-Hiickel theory for each concentration. Compare these experimental and theoretical values. 

Extension of the Debye-Hiickel Theory.—In the calculation of the electrical density in the vicinity of an ion (p. 82), it was assumed that ZiGp/kT was negligible in comparison with unity, so that all terms beyond the first in the exponential series could be neglected. According to calculations made by Muller (1927), the neglect of the additional terms is justifiable provided that... 

The interpretation of titration curves of peptides and proteins can be quite tricky. In addition to the number of groups that may be involved, their pAa values can be perturbed by several factors. For example, when charged groups are in close proximity and when salts are present, pA, values are influenced by electrostatic effects. Titration thus gives apparent pAa values and the intrinsic values have to be computed by applying a correction factor based on the Debye-Hiickel theory ... 

The momentary association of simple ions is a well-known phenomenon that has been treated in a number of ways. For example, the ion association constant of Bjerrum has received much experimental support. However, the association of simple electrolytes is considered to be shortlived and has been included in the Debye-Hiickel electrostatic theory as correction constants to the concentration. On the contrary, the hydration of the ions may be long-lived. This may be accounted for by considering additionally the ionic interaction ... 

In addition to hydrochloric acid, the results for which have just been described in detail, the method utilizing concentration cells with transference has been used in obtaining the activity coefficients of potassium chloride,17 sodium chloride,18 silver nitrate,10 and calcium chloride.17 The resulting activity coefficients, /, and comparisons with equation (45), Chapter 7, of the Debye-Hiickel theory,... 

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]... 

The early conductance theories given by Debye and Hiickel in 1926, Onsager in 1927 and Fuoss and Onsager in 1932 used a model which assumed all the postulates of the Debye-Hiickel theory (see Section 10.3). The factors which have to be considered in addition are the effects of the asymmetric ionic atmosphere, i.e. relaxation and electrophoresis, and viscous drag due to the frictional effects of the solvent on the movement of an ion under an applied external field. These effects result in a decreased ionic velocity and decreased ionic molar conductivity and become greater as the concentration increases. 

The properties of the ions and the solvent which are ignored are similar to those ignored in the Debye-Hiickel treatment. These are very important properties at the microscopic level, but it would be a thankless task to try to incorporate them into the treatment used in the 1957 equation. Furthermore, Stokes Law is used in the equations describing the movement of the ions. This law applies to the motion of a macroscopic sphere through a structureless continuous medium. But the ions are microscopic species and the solvent is not structureless and use of Stokes Law is approximate in the extreme. Likewise, the equations describing the motion also involve the viscosity which is a macroscopic property of the solvent and does not include any of the important microscopic details of the solvent structure. The macroscopic relative permittivity also appears in the equation. This is certainly not valid in the vicinity of an ion because the intense electrical field due to an ion will cause dielectric saturation of the solvent immediately around the ion. In addition, alteration of the solvent stmcture by the ion is an important feature of electrolyte solutions (see Section 13.16). However, solvation is ignored. As in the Debye-Hiickel treatment the physical meaning of the distance of closest approach, i.e. a is also open to debate. 

Note that, compared with the counterpart equations without the hydration correction, the factor D appears in various terms. In the equations for molal activity coefficients, there are two additional terms, one containing In D, the other containing the Debye-Hiickel function f. 

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