Chapman equations

Several features of the behavior of the Gouy-Chapman equations are illus-... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Nakagaki1U) has given a theoretical treatment of the electrostatic interactions by using the Gouy-Chapman equation for the relation between the surface charge density oe and surface potential /. The experimental data for (Lys)n agrees very well with the theoretical curve obtained. 

Soils containing polyvalent cations having high valence and high electrolyte concentration have a high conductivity, whereas the soils containing monovalent cations, such as sodium, have a low k. Distilled water at the extreme end of the spectrum is free of electrolytes. In the Gouy-Chapman equation, the electrolyte concentration na would be 0. The denominator, therefore, would go to 0 and the T value to infinity. 

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The surface charge density of the diffuse part of the double layer is given by the Gouy-Chapman equation ... 

Equation (1.35) is known as the Debye-Hiickel or Gui-Chapman equation for the equilibrium double layer potential. In terms of the original variable x (1.34), (1.35) suggest e1/2(r(j) is the correct scale of ip variation, that is, the correct scale for the thickness of the electric double layer. At the same time, it is observed from (1.32) that for N 1 the appropriate scale depends on N, shrinking to zero when N — oo (ipm — — oo). This illustrates the previously made statement concerning the meaningfulness of the presented interpretation of relectric potential

(—oo) — 0, < (oo) — —oo). 

The applicability of Equation 11 improves when the rate of aggregation is not too slow, e.g., when the surface potentials (calculated from the modified Gouy Chapman equation (see Refs. 28 and 29)) are not too high. These are the only cases of interest for quantitative comparison with experiment, since detection becomes impossible when the rate of aggregation is too slow. We point out below how we estimate rates for other cases. 

Although each SCM shares certain common features the formulation of the adsorption planes is different for each SCM. In the DDLM the relationship between surface charge, diffuse-layer potential, d, is calculated via the Gouy-Chapman equation (Table 5.1), while in the CCM a linear relationship between surface potential, s, is assumed by assigning a constant value for the inner-layer capacitance, kBoth models assume that the adsorbed species form inner-sphere complexes with surface hydroxyls. The TLM in its original... 

Finally, it may be noted that, since lim, -, p = 0, o-Ac= may be related to through Eq, [3] (which reduces to the Gouy—Chapman equation when p = 0). Substituting this relation into [9] ... 

Drzymala and coworkers proposed the method of the calculation of surface hydroxyl group complexion constants, where the concentrations of the group from Eqs. (25) and (26) are calculated in the similar to Schindler s and Gamsjager way, whereas the surface potential value is determined from the Gouy-Chapman equation [111]. The logarithm of such obtained values of the constants, differs from that of obtained by Schindler s method by a few tenths of the pK unit. 

Complementary is the method of complexation constant calculations based on the adsorption measurements of the 1 1 background electrolyte. The density of this adsorption consists of = SOH An+ or = SO Ct+ complex adsorption and a part, connected with the compensation of the surface charge in the diffuse layer of edl. To estimate the densities, the ions adsorbed in IHP layer, Sprycha assumed the background electrolyte ion density located in the diffuse layer of edl equals to the diffuse layer charge that may be calculated from Gouy-Chapman equation, when potential value is known. Then, the density of the ions that form surface complexes will be equal to ... 

If all ions are assumed to have the same charge q, (i.e., a symmetric salt) and the colloid is treated as a flat and infinitely large surface in contact with an infinite salt reservoir, the PB equations can be solved and the solution is denoted the Gouy-Chapman equation. Under these conditions Eq. (4) can be rewritten as... 

The linearization of the PB equation is often called the Debye-Hiickel approximation and it is valid when qfo/kT < 1. At room temperature this corresponds to surface potentials, 0o, below 25 mV. In the case of flat surfaces and if symmetry is considered as in the Gouy-Chapman equation... 

Strategy. Let us assume that the Earth s stratosphere is a large homogeneous compartment and that the flows of O2,0, and O3 are given by the four Chapman equations. The concentration of M (N2 and 02) is sufficiently high so that it is virtually a constant. To solve this problem, let us first set up the equations for the steady-state concentrations of O and 03 in other words, we will set up the equations for the rates of formation of O and O3 and set these rates equal to zero. Using these two expressions, we will then calculate the value of the 03 to... 

What is the ozone concentration at 60 km altitude You will need to know that the rate constants for the four Chapman equations vary with temperature as follows ... 

Calculation of Mean Free Path The Chapman equation permits calculation of mean free path from viscosity measurements. According to this equation... 

Example. At 27 C and 1 atm pressure, the coefficient of viscosity of nitrogen gas is 178 pP (i.e., micropoise). Calculate, (a) the mean free path X, and (b) the collision diameter o of nitrogen molecule using the Chapman equation. 

The surface potential of a bilayer is a result of having charged lipids in this bilayer (for reviews see References 63 and 66). In fluid phase bilayers, rapid translational diffusion of the lipids allows the surface charge associated with the lipids to be considered a smeared charge and the electrostatic potential at the surface of the bilayer, J o, is well described by the Gouy-Chapman equation ... 

Figure 34. Determination of micellar charge from equilibrium and kinetic measurements. The decrement of micellar charge as a function of sodium dodecyl sulfate added to Brij 58 micelles was calculated from the pK shift according to Gouy-Chapman equation (/ = lOm/W) (A) or from the second-order rate constant of protonation using Debye s equation (Eigen etal., 1964) for rates measured in the presence of ionic screening (O) at/ = 10mM,or from rates extrapolated to / = 0 ( ) (Gutman et al., 1981a).

One basic difficulty with the Gouy-Chapman Theory in systems involving an impenetrable flat surface or electrode is that, since the ions have finite size, the distance of closest approach of their centers to the surface is finite. Thus, the potential which appears in the Gouy-Chapman equation is not equal to the surface potential A(0), but is the potential in the plane of closest approach of the counterions to the surface. 

Fig. 8.4-14 Electrostatic potential as a function of the distance. The potential rises from the value at Z) = 0 within the adsorbed layer to a maximum in the Helmholtz layer followed by a linear decrease to the Stem potential. The Gnoy-Chapman equation describes this decrease in the double layer...

In one respect Derjag.uin s method is superior-to Levine and. Dube s. It is possible to. apply the principle of Der-j a g u i n s method to the complete Gouy-Chapman equation, whereas in Levine s and Dube s treatment it is unavoidable to introduce the linear approximation of Debye and Huekel. The theory for large spherical particles, surrounded by a thin double layer may therefore be made almost as exact as that of flat plates, but in the case of small particles with an extended double layer the situation is less favourable. 

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