Debye-Huckel approximations

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

What are the assumptions that are needed to obtain the linearized Poisson-Boltzmann (LPB) equation from the Poisson-Boltzmann equation, and under what conditions would you expect the LPB equation to be sufficiently accurate What is the relation between the Debye-Huckel approximation and the LPB equation ... 

Finally, Fei st describes electrostatic interactions and in Debye-Huckel approximation is given by [23, 26] ... 

Correct measured values for liquid junction potentials using the Henderson formalism and calculate ion activities according to the Debye-Huckel approximation. 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

Debye-Huckel approximation — In calculating the potential distribution around a charge in a solution of a strong -> electrolyte, - Debye and -> Hiickel made the assumption that the electrical energy is small compared to the thermal energy ( zjei (kT), and they solved the -> Poisson-Boltzmann equation V2f = - jT- gc° eexp( y) by expanding the exponential... 

See also -> Huckel equation of electrophoretic mobility, -> Debye-Huckel approximation, -> Debye-Huckel length, -> Debye-Huckel limiting law, -> Debye-Huckel-Onsager theory, -> Debye-Huckel parameter. 

Figure 3.13. Potential distribution in a spherical diffuse double layer. Symmetrical electrolyte. Dashed curves Debye-Huckel approximation. (Back-scaling in terms of concentrations. radii and valencies can be done by using 13.5.8] for K and a (in nm) = 0.3042 Ka / z-Jc with c in M at 25 C.)...

Debye-Falkenhagen effect 1.5.60,1.6.6c, 4.111 Debye-Huckel approximation 1.5.19(intr.)... 

As an illustrative example taken from Russel et al. (1989), let us consider a 0.01 molar solution of sodium chloride in contact with a surface charged at a density of 5 x 10 negative charges per square meter at room temperature, 298°K. Equation (2-52) gives /c = 3 nm. The dimensionless surface potential exfJkrtT. obtained from Eq. (2-45), is —5.21, and Eqs. (2-46) and (2-49) give respectively the exact and the Debye-Huckel approximations for the potential as a function of distance from the surface. The results are plotted in Fig. 2-13. Note that since — s/ ks T > 1, the Debye-Huckel approximation is... 

P. C. Meier, Two-Parameter Debye-Huckel Approximation for the Evaluation of Mean Activity Coefficients of 109 Electrolytes, Anal. Chim. Acta, 136... 

A re-interpretation of the results was attempted, by modelling pH and solution speciations with the help of the GEMS-PSI code [2004KUL/BER] [2005CUR/KUL], using the extended Debye-Huckel approximation. The hydrolysis model selected in this review was applied, assuming simultaneous formation of one or more carbonate species. The main results are shown in Figure A-32 and can be summarized as follows. 

Using the concentration and velocity distributions calculated in Section 6.5, we obtain the total current by quadrature. We do not carry out the integrations here but merely note that within the Debye-Huckel approximation the current corresponding to the constant surface potential solution represented by Eqs. (6.5.23) and (6.5.24) is... 

Thus, in the considered case, the Debye thickness is the distance from the charged plane, at which the potential decreases by a factor of e. In fact, near the surface, the potential is not necessarily small, so the Debye-Huckel approximation can be violated, and (j) can fall faster than suggested by (7.68). A double layer in which (j) changes from to 0 as described by Eq. (7.66) or by Eq. (7.67) is called a diffusion layer. 

Thus, q is associated with the distribution of rj) over the double layer. In the case of a very thin layer, r should be understood as a direction normal to the surface. In the Debye-Huckel approximation, we have ... 

The electrophoresis retardation is motion of ions in the double layer in direction opposite to particles motion. Due to forces of viscous friction, the ions cause the electroosmotic motion of liquid, which retards the particle s motion. Following the approach presented in [48], consider the electrophoresis motion of a particle, assuming that the double layer remains spherical during the motion, and the potential of the particle s surface is small enough, so Debye-Huckel approximation is valid. The motion is supposed to be inertialess. Introduce a coordinate system moving with the particle s velocity U so that in the chosen system of coordinates the particle is motionless, and the flow velocity at infinity is equal to - 17 (Fig. 9.2). 

The Debye-Huckel approximation gives the potential distribution in the electrical double layer exp(—x/2d). Then we have at the symmetry axis... 

To solve for the potential, we consider the limit of high salt concentration so that the fixed charges are well screened. We assume that the potential is small (more precisely that it does not change much from the plates to the center) and linearize Eq. (5.77). This approximation is called the Debye-Huckel approximation. We then find... 

The value of O at the distance r = 1 / c, where the ionic atmosphere is most densely populated, gives an estimate of the validity of the Debye-Huckel approximation. Neglecting factors of the order of unity, it turns out that q = [ zeYlsrkT) K < 1 hence at 25°C, / c) < 5.10 v/z. Thus the linearised Poisson-Boltzmann relation underestimates the electrostatic interactions in polyvalent electrolytes and even for (1-1) salts in solvents of low dielectric constant. 

On the other hand, since Bjerrum s treatment is an extension of Debye and Hiickel s, it is natural that when the electrostatic interactions are small enough to make the Debye-Huckel approximation valid (r > d) there is no need to invoke ionic association. 

The Debye-Huckel approximation may be used if the surface potential is small ... 

For a spherical double layer, the solution using the Debye-Huckel approximation, will be ... 

Debye-Huckel Approximation In some situations where the zeta potential is small (i.e., < 25 mV), the hyperbolic function in Eq. 16 can be approximated as sinh(ze //A b7 ze lk, T, which is called the Debye-Huckel approximation. Equation 16 then becomes... 

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