The Chapman Model

The formation of the stratospheric ozone layer can be understood most simply on the basis of a reaction model composed of a minimum set of four elementary processes (a) the dissociation of oxygen molecules by solar radiation in the wavelength region 180-240 nm (b) the attachment of oxygen atoms to molecular oxygen, leading to the formation of ozone (c) the photodissociation of ozone in the Hartley band between 200 and 300 nm and (d) the destruction of ozone by its reaction with oxygen atoms. The reactions may be written 

As a check for the correct application of the stoichiometry factors, we multiply the second equation by two, and third by three, and obtain after summing 

Integration then yields the mass balance equation 

Fortunately, a number of simplifications are possible. First, it may be noted that the O-atom concentration approaches the steady state much faster than the ozone concentration. At 50 km altitude, the time constant for the adjustment to steady state of n, is r0= l/fcbn2nM = 20s, and the value decreases as one goes toward lower altitudes. The time constant for the approach to steady state of ozone, in contrast, is much longer, and it increases with decreasing altitude (see below). It is thus reasonable to assume that oxygen atoms are always in steady state, that is, dn,/dt = 0. In addition, it turns out that the second and third terms in Eq. (3-1) are dominant compared with the other two. Accordingly, one has approximately 

The first and last terms of Eq. (3-1) cannot be neglected, of course, because they are ultimately responsible for the production and the removal of the sum of oxygen atoms and ozone. This fact may be taken into account by summing Eqs. (3-1) and (3-3), and by using the sum equation in conjunction 

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This simple oxygen-only mechanism consistently overestimates the O3 concentration in the stratosphere as compared to measured values. This implies that there must be a mechanism for ozone destruction that the Chapman model does not account for. A series of catalytic ozone-destroying reactions causes the discrepancy. Shown below is an ozone-destroying mechanism with NO/NO2 serving as a catalyst ... 

The rates of reactions (149)-(152) vary with altitude. The rate constants of reactions (149) and (151) are determined by the solar flux at a given altitude, and the rate constants of the other reactions are determined by the temperature at that altitude. However, precise solar data obtained from rocket experiments and better kinetic data for reactions (150)-( 152), coupled with recent meteorological analysis, have shown that the Chapman model was seriously flawed. The concentrations predicted by the model were essentially too high. Something else was affecting the ozone. 

Ozone production does not require low pressure in the Chapman model. Changing the oxygen pressure to 100 Torr gives 0.025% ozone after 4 h but the increased collision rate reduces atomic oxygen to 1/5 th the value at 10 Torr. However, a pressure increase may require the inclusion in the mechanism of the step O3-EM 0-E02-EM. This would reduce ozone production. 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

Fig. 20.9 Experimental capacitance-potential curve for O-OOI m KCl and calculated curve using the Gouy-Chapman model. The experimental curve and the theoretical curve agree at potentials (us R.H.E.) near the p.z.c. Note the constant capacitance of 17 x 10 F m at negative potentials (after Bockris and Drazic )...

The Stern model (1924) may be regarded as a synthesis of the Helmholz model of a layer of ions in contact with the electrode (Fig. 20.2) and the Gouy-Chapman diffuse model (Fig. 20.10), and it follows that the net charge density on the solution side of the interphase is now given by... 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

The physical meaning of the g" (ion) potential depends on the accepted model of ionic double layer. The proposed models correspond to the Gouy Chapman diffuse layer, with or without allowance for the Stern modification and/or the penetration of small counterions above the plane of the ionic heads of the adsorbed large ions [17,18]. The presence of adsorbed Langmuir monolayers may induce very high changes of the surface potential of water. For example. A/" shifts attaining ca. —0.9 (hexadecylamine hydrochloride), and ca. -bl.OV (perfluorodecanoic acid) have been observed [68]. 

The simplest model for the ionic distribution at liquid-liquid interfaces is the Verwey-Niessen model [10], which consists of two Gouy-Chapman space-charge layers back to... 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

We recently synthesized several reasonably surface-active crown-ether-based ionophores. This type of ionophore in fact gave Nernstian slopes for corresponding primary ions with its ionophore of one order or less concentrations than the lowest allowable concentrations for Nernstian slopes with conventional counterpart ionophores. Furthermore, the detection limit was relatively improved with increased offset potentials due to the efficient and increased primary ion uptake into the vicinity of the membrane interface by surfactant ionophores selectively located there. These results were again well explained by the derived model essentially based on the Gouy-Chapman theory. Just like other interfacial phenomena, the surface and bulk phase of the ionophore incorporated liquid membrane may naturally be speculated to be more or less different. The SHG results presented here is one of strong evidence indicating that this is in fact true rather than speculation. 

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