Some Model Calculations

In this subsection we present the work of Menou et al., 2003 [229] who simulated the motion of a thin, homogeneous layer of hydrostatically balanced, inviscid fluid 

In dimensionless form, shallow-water equations become functions of the Rossby (Ro) and Burger (Bu) numbers  

H are the characteristic velocity, length and layer thickness scales. L/j is the Rossby deformation radius. The Rossby number measures the importance of rotation on the flow, the Burger number measures the stratification in the atmosphere via the Brunt-Vaisala frequency. As it is known, the atmospheric structure in the giant planets in the solar system is in bands. This structure evolves from shallow-water turbulence. Two dimensional turbulence is characterized by an inverse energy cascade—this means a transfer from small to large scales. 

A simple atmospheric model must take into account several effects The photodissociation rate J(z) at height z in the atmosphere is 

Fy is the input flux from the host star, Xy the optical depth which depends on the opacity (cm ) and Oy is the photodissociation cross-section. AU these quantities depend on the frequency v. Photons may be absorbed and Rayleigh scattering could also be important. Both processes provide a shielding of an atmosphere from UV radiation. 

---

The consequences on the magnitude of the secondary a-deuterium KIE of coupling the motion of the nontransferring a-hydrogen into the reaction coordinate motion, as suggested by Kurz and Frieden, was investigated in some model calculations by Huskey and Schowen (1983). They used two different models to calculate the secondary isotope effects for the hydride transfer reaction (45). 

Numerous X-ray investigations have unravelled the solid state structure of contact and solvent-separated ion pairs. It was therefore considered to be of interest to evaluate also the potential of solid state NMR as a tool for the investigation of this structural problem. In addition to the study of chemical shifts discussed above (Section II.B), the quadrupole coupling constant of the nuclide Li, x( Li), was expected to be an ideal sensor for the bonding situation around the lithium cation because, due to its dependence on the electric field gradient, the quadrupolar interaction for this spin-3/2 nucleus is strongly influenced by local symmetry, as exemplified in Section II.C.3. This is also shown with some model calculations in Section ILF. 

Reconstruction of Au(lll) is observed in STM images as double rows separated from each other by 6.3 nm [335]. Some model calculations have been performed [362] to show that the energy difference of the reconstructed and unreconstructed Au(lll) is small. The effect of Triton X-100 on the reconstruction process of Au(lll) surface has been studied in chloride media [363] applying CV and double potential-step chronocoulometry. It has been found that adsorption of Triton X-100 stabilizes the reconstructed face of Au(lll). Hobara etal. [364] have used in situ STM to study reconstruction of Au(lll), following reductive desorption of 2-mercaptoethanesulfonic acid SAMs. 

Figure 12.9 shows some model-calculated percent changes in total column ozone due to a HSCT fleet that was projected in 2015 assuming the emission goal of El no = 5 g of NOz/kg of fuel was met (Stolarski et al., 1995). These calculations compare the change in 03 due to this fleet compared to a completely subsonic fleet in that year using the three different models for which predicted altitude changes were shown in Fig. 

The main questions which arise with respect to the Glueckauf model discussed above concern the uniqueness and the physical justification of the postulated local charge variation represented by Eq. (44). With reference to the former question, some model calculations of SN for f(CFL) distributions of different shapes and widths may be noted 122,123). it was shown that the magnitude and form of the deviation from EVM behaviour for a given B/A ratio is determined largely by the shape of the f(CFL) function near the lower limit A. The precise form of f(CFL) in the higher CFL region appears to have a relatively minor effect on the conformity of SN to the power law of Eq. (46) or the value of q. 

Although the low field effects on chemical reactions through radical pairs had been explained by the LCM, Timmel et al. [14] proposed that the so-called low field effects arose also fi om coherent superpositions of degenerate electron-nuclear spin states in a radical pair in zero field. They made some model calculations for their mechanism. At first, let us consider the case of a radical pair with a single spin-1/2 nucleus, e.g., a proton. When the exchange term is not included (J= 0 J), its spin Hamiltonian (H) can be expressed from Eq. (3-3) as... 

The experimental data are shown In Figure 11a, together with a dotted line, extrapolating the reaction controlled regime to zero time. On the other hand some model calculations have been performed under the assumption that, at the beginning of the experiment, the reaction rate is in the diffusion controlled regime. Then only a restricted spherical shell close to the bead surface Is participating In the substrate conversion. If superimposed on the reaction a time dependent catalyst deactivation occurs, the reactive shell should move to the center of the catalyst particle where active cells are available to substitute for the deactivated cells closer to the surface. 

Photon absorption provides another route for obtaining specific rate constants from highly excited molecules. Making some assumptions about collisional energy transfer, one may use the collision frequency as a clock in the measurement of the competition between photoreaction and collisional stabilization. Alternatively, one may make some model calculations for the specific rate constants for photoreaction and for collisional energy transfer. From these one may compute experimental quantities like photolysis quantum yields as a function of pressure and test them for consistency with experiments. This procedure has been used in a detailed study of NO2 photolysis at wavelengths between 313 and 416 nm (predissociation threshold at 397.9nm) and N2 pressures between 0 and 1000atm. The... 

Some model calculations using Eqs. (18)-(20) are summarised in Fig. 4. The coefficient a was set zero and b was again adjusted such that the concentration lO" mold corresponds to a surface pressure of 30 mN/m. For coi = C(>2 we get the curve for the Langmuir model (curve 2 - dotted line). 

Some model calculations will provide an insight into polydispersity effects. Only the chain length polydispersity is taken into consideration (in a first approximation). The segment-molar distribution function is assumed to be of the type... 

A possible model for the bending moles of the transition state is the bending vibration of the symmetrical ion HF7, which has v = 1225 cm Since, in the transition state, the proton is still close to two centres, it seems likely that in general the bending frequency will be at least as great as that in a normal molecule, and we shall see later in this chapter that this expectation is borne out by some model calculations. A further contribution to Eq comes from the symmetrical stretching mode... 

Surprisingly, somewhat larger A values result for the more remote hydrogen atoms (Tables 18 and 22), According to some model calculations focused inter alia on the free Sm(III) ion, extremely low Fermi contact A o and moderately weak magnetic dipolar contributions, A P, of A so have in fact been predicted for this particular lanthanoid ion (see Table 7). 

Some model calculations made by Parsons [11] for the reduction of anions and cations on mercury from NaF show how dramatic the deviations can be (Fig. 5.10). 

Subsequently, the theory is applied to stepwise fi actionation using the crossfractionation procedure. After some model calculations to study the influence of different operative fractionation parameters on the fractionation efficiency, the theoretical results will be again compared with experimental data for the styrene-butadiene copolymer system in two different solvent systems, namely cyclohexane + isooctane and benzene + methyl ethyl ketone [75]. 

---