Debye-Huckel equation corrections

This is the electrostatic energy arising from ions approaching within a of each other. When subtracted from the free energy functional above the corrected Debye Huckel equation becomes... 

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. 

The Debye-Huckel equation (and other empirical expressions that correct measured concentrations to activities) fails to account for specific ion pairing and com-plexation in solution, which in some salt solutions may contribute more to the inequality between concentration and activity than the nonspecific electrostatic interactions modeled by the equation. Ion pairing or complexation is likely to become significant in solutions with any of the foUowing characteristics ... 

In most soil solutions, the ionic strength I is low (<0.01), so that the extended Debye-Huckel equation is applicable for the correction of ionic concentrations to the more thermodynamically meaningful activities. Typically, conductivity measurements are used to estimate the ionic strength, a much less laborious procedure than measurement of each cation and anion present in solution. More problematic, however, is the detection and measurement of complexing anions and molecules (ligands). As will be shown later in this chapter, their presence can result in activities of metal ions in soil solution being much lower than measured concentrations would suggest. 

This tendency to decrease the mobility of the Ba is related directly to the ionic strength of the solution. By incorporating the Debye-Huckel equation, the following correction term is obtained ... 

The correction factor C may be determined from pH measurements on appropriate acid solutions, or calculated using the Debye-Huckel equation. For aqueous solutions at 25°C and ionic strength of 0.10, the correction factor is often taken to be -0.10 (calculated from the Debye-Huckel equation). Experimentally measured values are similar. For the calculation of [OH-], the concentration-product value of Kw must be used at 25°C and I = 0.10, pcJ(w = 13.75. 

The thermodynamic parameters for the formation of several metal-cyanide complexes, among others those of Ni(CN)4 , have been determined using pH-metric and calorimetric methods at 10, 25 and 40°C. In case of nickel(II), the thermodynamic data were determined by titration of Ni(C104)2 solutions with NaCN solutions. The ionic strength of the solutions were 1 < 0.02 M in all cases. The Debye-Huckel equation, related to the SIT model, was used to correct the formation constants to thermodynamic constants valid at 7 = 0. Since previous experiments indicated that the dependence of A,77° in the ionic strength in dilute aqueous solutions is small compared to the experimental error, the measured heats of reaction (A,77 = - 189.1 kJ mol at 10°C A,77 ,= -183.7 kJ mol at 40 C) were taken to be valid at 7 = 0, but the uncertainties were estimated in this review as 2.0 kJ moT. From the values of A,77 , as a function of temperature, average A,C° values were calculated. 

NB Apparent pKa values were determined by titration in eitiier aqueous DMSO (30-80 wt%) solutions or in aqueous metiianol (10-50 wt%) solutions. Ionic strength effects were corrected with Davies modification of the Debye-Huckel equation. Wei t percent compositions were converted to mole fraction and plots (often exhibiting traces of curvature) of apparent pX were extrapolated to 100% water by linear regression analysis. The following values were reported for tiie pJCa value in water (pX )-... 

The same functional dependence of b appears in the corrections to the Debye-Huckel equation obtained by more exact statistical mechanical calc ations. Equation 5.3.14 may be compared to eqn. 5.3.10, the difference between them being the factor Ijb, which does not appear in... 

Dissociation rates were obtained at high acidity where kn is fast and were corrected for ionic strength using a Debye-Huckel equation. The results show that the cryptate selectivity results mainly from kh and that the transition state for the reaction has little interaction between the metal and the cryptand to differentiate between metals. Rates kt increase with increasing cavity size. The thermodynamics of cryptand formation in water and methanol have also been used to calculate the free energy and enthalpy of transfer of the free ions between the two media. 

For a dilute electrolyte solution, the correction can be calculated using the extended Debye-Huckel equation ... 

Temperature Effects. The temperature range for which this model was assumed to be valid was 0°C through 40°C, which is a range covering most natural surface water systems (28). Equilibrium constants were adjusted for temperature effects using the Van t Hoff relation whenever appropriate enthalpy data was available (23, 24, 25). Activity and osmotic coefficients were temperature corrected by empirical equations describing the temperature dependence of the Debye-Huckel parameters of equations 20 and 21. These equations, obtained by curve-fitting published data (13), were... 

Banks [34BAN] made careful measurements of the conductivity of zinc selenate solutions at 298.15 K. The concentration range used was approximately 2 x 10 to 1 X 10 M. On the assumption that only ZnSe04(aq) was formed the data were evaluated by an iterative procedure in which the inter-ionic attraction was corrected for using the Debye-Huckel (activity coefficient) and Onsager (ionic mobility) equations. The result for ... 

From this it can be seen that the correct form of the Debye-Huckel-Onsager equation to be used to compare with the experimental results is one in which stoichiometric concentrations appear. 

The correction factor,/, relates the actual mobility of a fully charged particle at the ionic strength under the experimental conditions to the absolute mobility. It takes ionic interactions into account and is derived for not-too-concentrated solutions by the theory of Debye-Huckel-Onsager using the model of an ionic cloud around a given central ion. It depends, in a complex way on the mean ionic activity coefficient. The resulting equation contains the solvent viscosity and dielectric constant in the denominator. In all cases, the factor is < 1. The actual mobility is always smaller than the absolute mobility. 

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 ... 

The ionic strength dependence of k is essentially a property of the rate law. Therefore, the ionic strength dependence seldom affords new mechanistic information unless the complete rate law cannot be determined. These equations more often are used to "correct" rate constants from one ionic strength to another for the purpose of rate constant comparison. Ionic strength effects have been used to estimate the charge at the active site in large biomolecules, but the theory is substantially changed because the size of the biomolecule violates basic assumptions of Debye-HUckel theory. 

This equation is valid for relatively thin double layers, k a 1 (k being the Debye-Huckel parameter). For high potentials an additional correction is required for the relaxation effect, which is similar to the situation described above. 

Both start out with the Debye-HUckel term which depends primarily on ionic strength but that term alone is applicable only to very dilute solutions. Guggenheim adds an essentially empirical first-order correction and this is sufficient for ionic strengths to about 1 or 2 molar. For more concentrated solutions, Pitzer has proposed a semi-theoretical equation which, however, has many parameters and all of these depend on temperature (22). 

In this context, a frankly empirical approach was adopted by earlier workers not yet blessed by Debye and Huckel s light. Solutions that obeyed Eq. (3.52) were characterized as ideal solutions since this equation applies to systems ofnoninteracting solute particles, i.e., ideal particles. Electrolytic solutions that do not obey the equation were said to be nonideal. In order to use an equation of the form of Eq. (3.52) to treat nonideal electrolytic solutions, an empirical correction factor was introduced by Lewis as a modifier of the concentration term ... 

This equation is often referred to as Ostwald s dilution law. It depends on the assumptions (a) that ionic conductivities have constant values, and are not dependent on the concentration and (b) that the ions in dilute solution behave as ideal solutes. Both of these assumptions proved to be mistaken, and were finally corrected in the interionic attraction theory of Debye and Huckel and the conductance equation of Onsager (see conductance of aqueous solutions, conductance equations). 

---