The Debye-Huckel Theory

There is no doubt that the mean distribution of positively and negatively charged ions around the central ion is spherically symmetrical, provided no additional forces are acting on the ions. The distribution simply represents the time-averaged effects of the mutual interaction and thermal motion of the ions. Therefore, the Laplacian turns into the following simple form  

The charge density in the volume element is equal to the excess charge in the volume element, which is equal to the sum of each ion density n, (the average number of i ions per unit volume) times the charge ZjC on the ion  

As mentioned above, there is no additional force applied to the system, and therefore the kinetic energy of ions owing to their thermal motion is expected to be much greater than their electrostatic energy, i.e., kT 1 

The symbol k is not only a shorthand symbol but also indicates a very important parameter concerning the distribution of ions around the central ion. The value of is proportional to ionic strength, and becomes zero as the solution approaches an infinite dilution. The linearized Poisson-Boltzmann equation (7.7) can be solved by the variable transform. The substitution y, = nj/, reduces (7.7) to the following form  

The differential equation is easily solved, and the solution becomes 

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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. 

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... 

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. 

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 ... 

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). 

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. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

The Debye-Huckel Theory The Finite-Ion-Size Model. If the approximation of the point charge is removed, the extended form of the Debye-Huckel law is obtained ... 

The scattering center is the metal ion, the Coulomb potential of which is screened in the manner of the Debye-Huckel theory of weak electrolytes. The screened potential has the form... 

The Debye-Huckel theory that we summarized in Chapter 11 is based on this assumption. In that chapter we gave the following equations that apply to limiting law behavior... 

From the Debye-Huckel theory for the potentiSI)(in the vicinity of an ion [96], Scatchard derived an expression for the effect of dielectric constant of the solvent ... 

The Debye-Huckel theory gives a way of dealing with non-ideality in solutions of electrolytes. The ideal free energy can be calculated, and the difference between the... 

The work terms wl (/ = r or p) are associated with the electrostatic work done when the reactants are brought together from infinity to a distance separated from rigid spheres. For ions of charges Zj and z2 in a medium with a dielectric constant D, w , i r or p, can be calculated on the basis of the Debye-Huckel theory (Equation 6.110). 

The activity coefficient yt of an ion depends on the ionic strength (I = ( )Zzfc , where zi is the charge number) according to the Debye-Huckel theory in the limit of low ionic strengths. As discussed in Section 1.2, this equation can be extended... 

In the Debye-Huckel theory, an ion in solution is treated as a conducting sphere. The distance of closest approach of two ions is a.4 The solution beyond a... 

The Debye-Huckel theory is accurate in solutions in which the interactions between ions are not too great (i.e., at low ionic strengths in solutions of monovalent ions in solvents with large dielectric constants). In Fig. 2, the predictions of Eq. (21) are compared with experimental data for some strong electrolytes with different ionic charges. 

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. 

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