Debye-Huckel theory equations

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 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. 

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... 

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

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

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. 

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. 

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... 

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 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... 

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... 

Begin with the Poisson equation but keep the e matrix inside the divergence operation V (eV0) = -4jrpext (see Fig. L3.24). The net electric-charge density pext at a given point depends on the magnitude of potential as in Debye-Huckel theory. As before in relation (L3.175),... 

The increase of the -> ionic strength (I) will influence the electrostatic interactions (-> Bronsteds salt effect, -+ Bronsted-Bjerrum equation) which can be taken into account by using the Debye-Huckel theory ... 

Debye-Huckel-Onsager equation Onsager equation, - Debye-Huckel-Onsager theory... 

To determine the spatial variation of a static electric field, one has to solve the Poisson equation for the appropriate charge distribution, subject to such boundary conditions as may pertain. The Poisson equation plays a central role in the Gouy-Chapman (- Gouy, - Chapman) electrical - double layer model and in the - Debye-Huckel theory of electrolyte solutions. In the first case the one-dimensional form of Eq. (2)... 

When the proper choice of the ponstants a and b are made, the function (Em° + Eext) should be constant within the limits of the extended Debye-Huckel theory. In calculating Em° the value of the equation for log y (Equation 6) which must be substituted into Equation 4 becomes... 

A Vital Step in the Debye-Huckel Theory of the Charge Distribution around Ions Linearization of the Boltzmann Equation... 

Does Mayer s theory of calculating the viriai coefficients in equations such as Eq. (3.165) (which gives rise directly to the expression for the osmotic pressure of an ionic solution and less directly to those for activity coefficients) really improve on the second and third generations of the Debye-Huckel theory—those involving, respectively, an accounting for ion size and for the water removed into long-lived hydration shells ... 

The Debye-Huckel-Onsager equation has been tested against a large body of accurate experimental data. A comparison of theory and experiment is shown in Fig. 4.93 and Table 4.21 for aqueous solutions of true eiectroiytes, i.e., substances that consisted of ions in their crystal lattices before they were dissolved in water. At very low concentrations (< 0.(X)3 N), the agreement between theory and experiment is very good. There is no doubt that the theoretical equation is a satisfactory expression for the limiting tangent to the experimentaiiy obtained/ versus curves. 

In addition to the short-range interactions between species in all solutions, long-range electrostatic interactions are found in electrolyte solutions. The deviation from ideal solution behavior caused by these electrostatic forces is usually calculated by some variation of the Debye-Huckel theory or the mean spherical approximation (MSA). These theories do not include terms for the short-range attractive and repulsive forces in the mixtures and are therefore usually combined with activity coefficient models or equations of state in order to describe the properties of electrolyte solutions. 

Heat Capacities and the Debye-Hfickel Theory.—By combining the general equation (44.46) with the expression for L2 [equation (44.39)3 derived from the Debye-Huckel theory, it is found that for a solution containing a single strong electrolyte,... 

Another well-known example of a medium effect in solution kinetics is the effect of ionic strength on reactions between ions [38, 39]. This is normally treated using the extended Debye-Huckel theory to estimate the activity coefficients in equation (7.10.1). The activity coefficient for species i is given by... 

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