Debye-Huckel theory with concentration

The first accurate calculation of the activity coefficient based on energetic effects of inter-ionic interactions in solvents was carried out by -> Debye and -> Huckel in 1923 by assuming that all the deviations from ideality at low concentrations of electrolyte were due to interionic interactions (- Debye-Huckel theory) with this it is possible to show that... 

It can be seen that y usually increases at high concentrations and, in some cases, becomes larger than 1, in disagreement with the Debye-Huckel theory. Some of the reasons proposed for the failure of the Debye-Huckel theory are ion pairing (loose association of oppositely charged ions in solution) and hydration of ions, with accompanying reduction of the amount of free solvent. 

Activity coefficients of H4[SiWi204o] obtained in 0.0004 to 0.04 M concentrations in queous solutions have also been reported11S. The results obtained agree with the Debye-Huckel theory for 1—4 electrolytes. Osmotic coefficients of 12-tungsto-silicic acid have also been reported116. ... 

In conclusion, therefore, it may be said that the treatment of the influence of ion-solvent interactions on ion-ion interactions has extended the range of concentration of an ionic solution which is accessible to theory. Whereas the finite-ion-size version of the Debye-Huckel theory did not permit theory to deal with solutions in a range of concentrations corresponding to those ofreal life, Eq. (3.130) advances theory into the range of practical concentrations. Apart from this numerical agreement with experiment, Eq. (3.130) unites two basic aspects of the situation inside an electrolytic solution, namely, ion-solvent interactions and ion-ion interactions. 

Electrolytes for which the concentration is less than lO Mcan usually be dealt with by the Debye-Huckel limiting law. Utilize the Debye-Huckel theory extended by allowance for ion size and also for removal of some of the active solvent into the ion s primary solvation shell to calculate the activity coefficient of 5 M NaCland 1M LaClj solutions (neglecting ion association or complexing). Take the total hydration number at the 5 M solution as 3 and at the 1 M solution as 5. Take r,- as 320 pm. 

In many calculations the hydrogen ion concentration is more accessible than the activity. For example, the electroneutrality condition is written in terms of concentrations rather than activities. Also, from stoichiometric considerations, the concentrations of solution components are often directly available. Therefore, the hydrogen ion concentration is most readily calculated from equilibrium constants written in terms of concentration. When a comparison of hydrogen ion concentrations with measured pH values is required (in calculation of equilibrium constants, for example), an estimate of the hydrogen ion activity coeflScient can be made by application of the Debye-Huckel theory if necessary, an estimate of liquid-junction potentials also can be made. Alternatively, the glass electrode can be calibrated with solutions of known hydrogen ion concentration and constant ionic strength. " ... 

Since both sides have high electrolyte strength, the activity coefficient term can be assumed identical for a given species in the two solutions, in accordance with the correlation based on modified Debye-Huckel theory proposed by Davies [28]. Therefore, the concentration term can replace the activity coefficient term and the condition for Donnan equilibrium becomes... 

In order to obtain activity coefficients or standard potentials from measurements on cells of the type shown in equation (1) it is necessary, as described in Chapters 8 and 10, to make some form of extrapolation. Furthermore in the test of the Debye-Huckel relations for these solutions it has been found that, because of the higher molecular weight of the non-aqueous solvents, the difference between the activity coefficient, f, and the rational coefficient, f, based on Raoult s law, cannot be neglected, even below a concentration of 0.1 normal, as it usually can with aqueous solutions. This difference was overlooked by the workers just mentioned, who found only partial agreement of their results with the Debye-Huckel theory. Their data have therefore been recomputed as follows. For the ethyl alcohol solutions, for instance, a reference solution with a molality, mu of 0.09501, was chosen. The relation... 

The water species involved in this reaction must be neutral (and not OH") because of the fact that the rate of uracil photohydrate formation is independent of NaCl concentration up to 1M, and is the same in unbuffered water as in 0.1M phosphate buffer. The rate constant for photohydrate formation in CU was also observed, in a series of runs all made in the same day with the same initial CU concentration, to be 0.0418 0.010 at NaCl concentrations of 0, 0.001M, 0.01M, 0.1 M, and 1M. The lack of salt effect is consonant, according to Debye-Huckel theory (3) with the reaction of a charged species (UH+) with an uncharged species, as written in Reaction f, and eliminates reaction between two charged species in the product-forming process. 

Comparison between thermodynamic values is generally made with standard state functions. To obtain the standard enthalpy of solution, A//g°in, it is necessary to extrapolate directly measured enthalpies of solution at finite concentrations to infinite dilution some form of the Debye-Huckel theory is generally used in this extrapolation (see sect. 2.5.2). 

Later, Falkenhagen and co-workers and Onsager and Fuoss established a method of calculating parameter A starting from the Debye-Huckel theory. However, the above equation is only valid for concentrations up to about 0.01 mol/L. According to the above equation the relative viscosity should always increase with concentration. However, experiments show non-monotonic behavior for several electrolytes such as most of the potassium halides, and several mbidium and cesium halides [12]. 

V Let us imagine the contact between two aqueous soiutions containing sodium chioride, each with a different concentration (Co, and Cp). The activity coefficients for the anions and cations in the same medium are set as equai. This is a vaiid approximation if appiying the Debye-HUckel theory, as it is a way of simpiifying the caicuiations without distorting the reasoning in any way. The ion activities in each of fhe fwo phases are denoted respectiveiy by ao,and ap. 

Classically, one treats phases of two components as ideal, regular, or real solutions. Usually, however, one concentrates for the non-ideal case only on solutions of salts by discussing the Debye-Huckel theory. Polymer science, in turn, adds the effect of different molecular sizes with the Hory-Huggins equation as of basic importance (Chap. 7). Considerable differences in size may, however, also occur in small molecules and their effects are hidden falsely in the activity coefficients of the general description. 

Electrostatic interactions give a large deviation from ideality in equilibrium properties of solutions containing low molecular weight electrolytes. This deviation was most successfully disposed of by the Debye-Huckel theory [2]. According to this theory, the ionic species are not distributed in solution in a random manner, but form an ionic atmosphere structure, and the thermodynamic properties such as the activity coefficient of solvent (or the osmotic coefficient), the mean activity coefficient of solute, and the heat of dilution, decrease linearly with the square root of the concentration, in conformity with experimental observations. 

This d is equal to the thickness of the ionic atmosphere in the Debye-Huckel theory for electrolytes for a 1 1 electrolyte, it increases with decreasing concentration ( ) and decreases with increasing concentrations. Thus the capacity of the double layer can also be written as... 

The derivation of the Debye— Hiickel equation for the activity coefficient is based on the linearized Boltzmann equation for electrostatic charge distribution around an ion. This limits the applicability of Eq. (57) to solutes with low surface potentials, which occurs for solution concentrations of monovalent ions of < 0.01 M. However, it is important to note that die method used for deriving activity coefficient equation (25) is based on rigorous thermodynamics and is not limited by the Debye—Huckel theory. If, for example, the Gouy—Chapman equation [22] was... 

The real behavior of systems is described by the activity coefficient y,. Instead of the concentration C of a dissolved species, one uses the activity a, = c y,. In the light of the Debye-Hiickel theory, y takes care of the electrostatic interactions of the ions. This is the main interaction for charged species in comparison with the smaller dipole and Van der Waals forces, which may be important in the case of uncharged species, but which are not included in the Debye-Huckel theory. The chemical potential p depends on the concentration according to Equation 1.37. 

Effects of the inert inorganic salts on the rate constants (k) for the reactions involving ionic reactants are generally explained in terms of the Debye-Huckel or extended Debye-Huckel theory. In actuality, the extended Debye-Huckel theory involves an empirical term, which makes the theory a semiempirical theory. However, there are many reports in which the effects of salts on k of such ionic reactions cannot be explained by the Debye-Huckel theory. For instance, pseudo-first-order rate constants (k bs) for the reaction of HO with acetyl salicylate ion (aspirin anion) show a fast increase at low salt concentration followed by a slow increase at high concentration of several salts. But the lowest salt concentration for each salt remains much higher than the limiting concentration (0.01 M for salts such as M+X ) above which the Debye-Huckel theory is no longer valid. These k bs values fit reasonably well to Equation 7.48... 

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