Debye—Huckel linearization

For low wall potentials the Debye-Huckel linearization holds and the excess charge distribution is... 

Poisson-Boltzmann equation (ie., Debye-Huckel linearization) is... 

The electric double layers formed at the microchannel walls do not overlap, and the Debye-Huckel linearization principle remains as valid. 

For further simplifications, one may note that for small values of the argument, it is possible to neglect higher order terms in an exponential series and approximate exp(jc) as I+x. This consideration, when applied to the potential distributions depicted by Eqs. (12a), (12b), forms the basis of the Debye-Huckel linearization principle [2], which effectively linearizes the pertinent exponential variation of ionic charge distribution for small values of e /k T. Under such approximations, Eq. (13) can be simplified as... 

The first term on the right-hand side of this equation is zero, since it is simply the sum of the electrical charge in solution, which must be zero for a neutral electrolyte solution. The third term is also zero for electrolytes with equal numbers of positive and negative ions, such as NaCl and MgSC>4. It would not be zero for asymmetric electrolytes such as CaCE. However, in the Debye-Huckel approach, all terms except the second are ignored for all ionic solutions. Substitution of the resulting expression into equation (7.20) gives the linear second-order differential equation... 

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 ... 

Equation 3 was obtained by combining the Nemst equation for the emf of Cell I with the equilibrium constant of the acidic dissociation of glycine. In Equations 2 and 3, E° is the standard emf of the cell in the respective solvent composition and these values were obtained from an earlier work (20). In Equation 4, /3 is the linear slope parameter for the plot of pK/ vs. I, a0 is the ion-size parameter, A and B are the Debye-Huckel constants on the molal scale (20) for the respective mixed solvent systems, and I is the ionic strength given by mi. 

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

Here we treat a planar plate surface immersed in an electrolyte solution of relative permittivity e,. and Debye-Huckel parameter k. We take x- and y-axes parallel to the plate surface and the z-axis perpendicular to the plate surface with its origin at the plate surface so that the region z>0 corresponds to the solution phase (Fig. 2.1). First we assume that the surface charge density a varies in the x-direction so that a is a function of x, that is, cr = cr(x). The electric potential ij/ is thus a function of x and z. We assume that the potential i/ (x, z) satisfies the following two-dimensional linearized Poison-Boltzmann equation, namely,... 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

The theoiy outlined above is a takeoff on the Debye Huckel theory of ionic solvation. In the electrochemistry literature it is known as the Gouy-Chapman theory. The Debye screening length is seen to depend linearly on ff and to decrease as (z+n+ +z zi ) /2 with increasing ionic densities. For a solution of monovalent salt, where z+ = z = 1 and = = n, this length is given by... 

In the search for medium complexes we first treated Na data at the lowest possible Na concentrations (self-medium solutions at low V,oi) to establish the medium independent constants and obtained log(3o,2,o = -0.30. This value was not allowed to vary in subsequent calculations. Then all Na data were treated. As previously found at high Vioi, two extremely minor resonances arising from an unprotonated linear trimer V. Oio " is discernable in such solutions. However, at the present high pH values, the amount is too low to make it possible to determine its formation constant. An approximate value of a monosodium complex, that fairly well explained these resonances, was used in the calculation in the search for all other complexes. All 35 experimental points were perfectly explained with non-sodium and sodium complexes of mono- and divanadates. An attempt to explain the data with extended Debye-Huckel expressions was not successful. 

NB Apparent pKa values were determined by titration in eitiier aqueous DMSO (30-80 wt%) solutions or in aqueous metiianol (10-50 wt%) solutions. Ionic strength effects were corrected with Davies modification of the Debye-Huckel equation. Wei t percent compositions were converted to mole fraction and plots (often exhibiting traces of curvature) of apparent pX were extrapolated to 100% water by linear regression analysis. The following values were reported for tiie pJCa value in water (pX )-... 

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... 

FIG. 5 (a, c) Surface charge density and (b, d) surface potential as functions of pH and pore size. (a. b) Linear potential-to-charge relation (Debye-Huckel) (c, d) nonlinear potential-to-charge relation (Poisson-Boltzmann). The charge regulation model used in the calculation was based on reaction set (34) with the following model parameters ... 

According to the Smoluchowski theory (Hunter 1981,1993), there is a linear relationship between the electrophoretic mobility and the potential U =At, where A is a constant for a thin EDL at Kfl 1 (where a denotes the particle radius and k is the Debye-Huckel parameter). For a thick EDL (Kfl< 1), e.g., at pH close to the isoelectric point (lEP), the equation with the Henry correction factor is more appropriate ... 

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. 

In view of the role one may presume for simple electrolyte - namely, its effect on the Debye-Huckel screening length - a dependence of Ksec on 1-1/2 was anticipated (25). However, the ionic strength was adjusted not by eluant modification, but by varying sample concentration, so the failure of the data for NaPSS eluted in pure H2O to conform to the aforementioned linear dependence is difficult to interpret. 

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