Rayleigh process

Figure 8. Example of apparent closed-system equilibrium fractionation, where Mo in solution is sorbed to Mn oxides (Barling and Anbar 2004). The 6 Mo values of the Mo remaining in solution during sorption follow die linear trends that are consistent widi closed-system equilibrium fractionation where isotopic equilibrium is continuously maintained between Mo in solution and diat sorbed to die Mn oxides. Three aqueous-solid pairs (shown widi tie lines) are consistent with this interpretation. The isotopic data cannot be ejqilained dirough a Rayleigh process, where die product of die reaction (sorbed Mo) is isolated from isotopic exchange widi aqueous Mo.

In contrast, reduction of Cr(Vt) in solution, followed by precipitation of hydrated Cr jOj, produces fractionation in that follows a Rayleigh process (Fig. 9). When plotted as a... 

Fig. 1.4 5 0 in a cloud vapour and condensate plotted as a function of a fraction of remaining vapour in a cloud for a Rayleigh process. The temperature of the cloud is shown on the lower axis. The increase in fractionation with decreasing temperature is taken into account (after Dansgaard 1964)... 

The intensity I of light scattered from an incident unpolarized beam of wavelength X and incident intensity Io by a single small particle of diameter d and refractive index n by the elastic, coherent Rayleigh process is given by... 

Chanical fractionation apparently cannot account for this difference. Kung et al. (1979) have found that N-isotope fractionations in both FTT and MiUer-Urey reactions are too small -l-3%o and + 10-12%o- The high SN of Cl and C2 chondrites could, in principle, be explained by mass fractionation in a Rayleigh process, involving loss of 99% of the N,. But this process would have to be driven to... 

If no liquid is kept within the cloud ( l = 0), these equations correspond to the classical Rayleigh model, and to a closed system if all liquid is kept. The isotopic composition of the precipitation can be calculated all along the condensation process, if the amount of water leaving the cloud at each step is known. The largest range of 5-variation corresponds to the immediate removal of the condensate (Rayleigh process), and there is a decrease of this range by about a factor of 2, when all liquid formed is kept in the cloud (Jouzel, 1986). 

Fig. 6. Carbon isotope fractionation in closed-system single-stage Rayleigh processes. Modified from Claypool and Kaplan (1974).

The foregoing is an equilibrium analysis, yet some transient effects are probably important to film resilience. Rayleigh [182] noted that surface freshly formed by some insult to the film would have a greater than equilibrium surface tension (note Fig. 11-15). A recent analysis [222] of the effect of surface elasticity on foam stability relates the nonequilibrium surfactant surface coverage to the foam retention time or time for a bubble to pass through a wet foam. The adsorption process is important in a new means of obtaining a foam by supplying vapor phase surfactants [223]. 

Perhaps the best known and most used optical spectroscopy which relies on the use of lasers is Raman spectroscopy. Because Raman spectroscopy is based on the inelastic scattering of photons, the signals are usually weak, and are often masked by fluorescence and/or Rayleigh scattering processes. The interest in usmg Raman for the vibrational characterization of surfaces arises from the fact that the teclmique can be used in situ under non-vacuum enviromnents, and also because it follows selection rules that complement those of IR spectroscopy. 

The resulting spectmm is illustrated in Figure 5.15, and Figure 5.16 shows in detail the processes involved in the first Stokes and anti-Stokes transitions and in the Rayleigh scattering. 

Figure 5.16 Raman and Rayleigh scattering processes involving virtual states Fq and Fj...

The mechanism for Stokes and anti-Stokes vibrational Raman transitions is analogous to that for rotational transitions, illustrated in Figure 5.16. As shown in Figure 6.3, intense monochromatic radiation may take the molecule from the u = 0 state to a virtual state Vq. Then it may return to u = 0 in a Rayleigh scattering process or to u = 1 in a Stokes Raman transition. Alternatively, it may go from the v = state to the virtual state Fj and return to V = (Rayleigh) or to u = 0 (Raman anti-Stokes). Flowever, in many molecules at normal... 

Natural Visibility Conditions visibility conditions attributable to Rayleigh scattering and aerosol associated with natural processes. 

Because exact solutions for skew isotropic plates are readily available, Ashton was able to get some exact solutions for anisotropic rectangular plates by the special identification process outlined in the preceding paragraph. Specifically, values for the center deflection of a uniformly loaded square plate are shown in Table 5-2. There, the exact solution is shown along with the Rayleigh-Ritz solution and the specially orthotropic solution. For the case already discussed where D22/D11 -1, (D.,2-i-2De6)/Di. = 1.5, and D.,g/Di., = D2 Di., =-.5, the exact solution is... 

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