Theory Debye-Huckel

The Poisson-Boltzmann equation for the electric potential, or equivalently distribution of ions, is nonlinear, and the solution of the equation requires numerical methods, except for a few special situations. However, tremendous simplification arises, without losing the major concepts of ion correlations, by linearizing the above nonlinear equation. Equation 3.13. 

Let us consider experimental conditions where the electrostatic interaction of the ions is relatively weak such that the electrical potential energy of an ion interacting with its cloud, ezaf, is small in comparison with the thermal energy ks T. Under these conditions, the exponential of Equation 3.13 can be expanded 

Due to the electroneutrality condition of Equation 3.1, the first term on the right-hand side of the above equation vanishes. By ignoring all the higher order terms inside the square brackets except the linear term in we get the linearized Poisson-Boltzmann equation, 

The linearized Poisson-Boltzmann equation is referred to as the Debye-Hiickel equation. The constant coefficient appearing in the above equation is one of the important parameters in the discussion of electrolyte solutions and polyelectrolytes. We shall describe its physical interpretation and experimental relevance below and in Section 3.1.3.1. 

Let us consider the electric potential around a reference ion i with charge ezi. Since the ion cloud is spherically symmetric on average. Equation 3.15 can be rewritten as 

The solubility reaction can also be written with respect to the formation of hydroxide ions, and the solubility constant for such a reaction is denoted (or often K (which is defined as the solubility product) again, and are 

The solubility of the generic phase given in reaction (2.13) can be such that rather than the free metal ion being the reaction product, other hydrolysis species may form depending on the reaction pH. The generic formula for such reactions can be derived by combination of reactions (2.5) and (2.13). The combined reaction is 

the solubility of the generic phase to form the hydrolysis species can be expressed in terms of the hydroxide ion, as defined by 

The Debye-Hiickel equation derives from a combination of the Poisson equation and a statistical-mechanical distribution formula (Debye and Hiickel, 1923). The Poisson equation is a general expression of the Coulomb law of force between charged bodies and can be written as 

Debye and Hiickel assumed the Boltzmann distribution law which states that since the electrical potential energy of a particular ion is zei//, the average local concentration of those ions at a point is 

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Fisher M and Levin Y 1993 Criticality in ionic fluids Debye Huckel Theory, Bjerrum and beyond Phys. Rev. Lett. 71 3826... 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Battery electrolytes are concentrated solutions of strong electrolytes and the Debye-Huckel theory of dilute solutions is only an approximation. Typical values for the resistivity of battery electrolytes range from about 1 ohmcm for sulfuric acid [7664-93-9] H2SO4, in lead—acid batteries and for potassium hydroxide [1310-58-3] KOH, in alkaline cells to about 100 ohmcm for organic electrolytes in lithium [7439-93-2] Li, batteries. 

It is shown that solute atoms differing in size from those of the solvent (carbon, in fact) can relieve hydrostatic stresses in a crystal and will thus migrate to the regions where they can relieve the most stress. As a result they will cluster round dislocations forming atmospheres similar to the ionic atmospheres of the Debye- Huckel theory ofeleeti oly tes. The conditions of formation and properties of these atmospheres are examined and the theory is applied to problems of precipitation, creep and the yield point."... 

Shortly after the formulation of the Debye-Huckel theory, a survey of the data on ionic mobilities from this point of view was made, extrapolating the values to infinite dilution.1 Table 4 gives values of Cl for atomic and molecular ions for 7 = 0°C and T2 = 18°C. 

In an aqueous solution at 25°C containing c moles/liter of a uni-univalont solute the value of dx according to Debye-HUckel theory is given by dx/kT —1.02 fc. 

If the activity coefficients are estimated from the Debye-Huckel theory in dilute regions of simple electrolyte systems, we have for aqueous solutions at 25 °C,... 

Improvements upon the Debye- Huckel Theory of Ionic Solutions The Manganese Dioxide Electrode in... 

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

Debye-Huckel theory assumes complete dissociation of electrolytes into solvated ions, and attributes ionic atmosphere formation to long-range physical forces of electrostatic attraction. The theory is adequate for describing the behaviour of strong 1 1 electrolytes in dilute aqueous solution but breaks down at higher concentrations. This is due to a chemical effect, namely that short-range electrostatic attraction occurs... 

This concept is due to Bjerrum, who in 1926 suggested that in simple electrolytes ions of the opposite charge could associate to form ion-pairs (Szwarc, 1965 Robinson Stokes, 1959). This concept of Bjerrum arose from problems with the Debye-Huckel theory, when the assumption that the electrostatic interaction was small compared with IcTwas not justified. 

When A > A the ions are free and the Debye-Huckel theory applies. When A < A the two ions tend to approach each other and form an ion-pair, and there is no contribution to the electrostatic energy from the interaction between an ion and its atmosphere. 

APPENDIX A DERIVATION OF THE MAIN EQUATION OF DEBYE-HUCKEL THEORY... 

APPENDIX A DERIVATION OE THE MAIN EQUATION OE DEBYE-HUCKEL THEORY 703... 

The question of the relationship between activity and concentration arises. Here the Debye-Huckel theory of activity coefficients, although valid only below 0.01 M, has proved to be most helpful, either for establishing an acid concentration from its H+ activity or for calculating H+ activity from its previously known acid concentration. 

Thus, a suitable refinement of the Debye-Huckel theory must provide a theoretical interpretation of the term CL Originally this term was qualitatively interpreted as a salting-out effect during solvation the ions... 

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Huckel 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 parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

De Broukere mean diameter, 18 135 Debt capital cost, 9 542 Debt ratio (DR), 9 541 Debt structure, 9 542-543 Deburring, surface, 9 597-598 Debutanizer, 10 614—615 Debye-Huckel theory, of electrolytes, 3 415 18... 

Bromley, L.A. "Approximate Individual Ion Values of 6 (or B) in Extended Debye-Huckel Theory for Uni-univalent Aqueous Solutions At 298.15 K," J.Chem. Thermo., 1972, 4, 669-73. 

Anderson, H. C., "Improvements upon the Debye-Huckel Theory... 

The theory proposed by Debye and Huckel dominated the study of aqueous electrolytes from around 1920 to near the end of the 1950 s. The Debye-Huckel theory was based on a model of electrolyte solutions in which the ions were treated as point charges (later as charged spheres), and the solvent was considered to be a homogeneous dielectric. Deviations from ideal behaviors were assumed to be due only to the long range electrostatic forces between ions. Refinements to include ion-ion pairing and ion... 

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

The Ky results of Sweeton, Mesmer and Baes (35) plotted in Fig. 2 were reported in 1974 and although they only extend to 300°C they may well be more accurate above this temperature than the experimental results of Fisher and Barnes(36), since, as mentioned, earlier, the Debye-Huckel theory may not give reliable extrapolations to infinite dilution at temperatures where water is highly compressible. While their work (35) involves extrapolation to infinite dilution as well as to higher temperatures it is very encouraging to note that their ACp at 300°C (-960 J K mol ) is of the magnitude expected on the basis of the NaCl studies referred to in Section 2. The conductance results of Sirota and Shviriaev (37) above 300°C also seem more consistent with the results of Sweeton, Mesmer and Baes (35), than with those of Fisher and Barnes (36). Marshall and Franck s recent representation of data up to 1000°C and 10,000 bars (38) predicts high temperature SVP results somewhat lower than those of Sirota and Shviriaev (37). 

Debye-Huckel theory phys chem A theory of the behavior of strong electrolytes, according to which each ion is surrounded by an ionic atmosphere of charges of the opposite sign whose behavior retards the movement of ions when a current is passed through the medium. do bT hik-ol, the-3-re ... 

Tanford examined the application of Debye-Huckel theory and found the theory not to be valid because the high charge density generatedby the closely spaced head groups leads to substantial charge neutralization by counter ions Alternatively, he equated the work of... 

Debye-Huckel Theory, As shown above the cations and anions in an aqueous solution are not uniformly distributed due to forces of interaction between them (ion-ion interaction). There is a statistical excess (over bulk concentration) of opposite charges around a given ion. Thus, ions in solution are surrounded by an ionic atmosphere of an opposite charge. The total charge in this ionic atmosphere is of opposite sign and equal to the charge of the particular ion. 

It has been pointed out above that electroosmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomena therefore the Helmholtz-von Smoluchowski equation based on the Debye-Huckel theory developed for electroosmosis applies to electrophoresis as well. In the case of electrophoresis, is the potential at the plane of share between a single ion and its counterions and the surrounding solution. 

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