Mass fractionation effects

Individual mass-transfer coefficient k, cm/s or ft/s k r, minimum coefficient for suspended particle (Fig. 21.6) k, effective internal coefficient [Eq. (21,65)] kc, average value over time tj, k, ky, in liquid phase and gas phase, respectively, based on mole-fraction differences, kg mol/m -s-unit mole fraction or lb mol/ft -h-unit mole fraction Effective mass-transfer coefficient in one-way diffusion kc, cm/s or ft/s k, in gas phase, kg mol/m -s-unit mole fraction or lb mol/ft -h-unit mole fraction also new values in Example 21.5 Length of pipe or tube, m or ft... 

Fig. 2. Electron drift velocities as a function of electric field for A, GaAs and B, Si The gradual saturation of curve B is characteristic of all indirect semiconductors. Curve A is characteristic of direct gap semiconductors and at low electric fields this curve has a steeper slope which reflects the larger electron mobiUty. The peak in curve A is the point at which a substantial fraction of the electrons have gained sufficient energy to populate the indirect L minimum which has a much larger electron-effective mass than the F minimum. Above 30 kV/cm (not shown) the drift velocity in Si exceeds that in...

The quantity G of the effective mixing mass flux is determined by the turbulent velocity fluctuations at the bubble-layer edge. The distance of the edge of the bubble layer from the wall is taken as the distance at which the size of the turbulent eddies is k times the average bubble diameter. Weisman and Pei have determined empirically that k equals 2.28. Only a fraction of the turbulent velocity fluctuations produced are assumed to be effective in reaching the wall. The effective velocity fluctuations are those in which the velocity exceeds the average velocity away from the wall produced by evaporation heat flux q"b. At the bubble layer-core interface, the effective mass flux to the wall is computed as... 

It is clear from figure 6 that the terrestrial data do not cluster about a single point but instead lie along a line of slope 0.5 on the three-isotope diagram, indicating isotopic variation due to mass-dependent fractionation. Since mass fractionation effects in Mg have not been observed in terrestrial materials [30,31], this distribution of observed isotope ratios must be due to fractionation in the ion probe. The physical process which produces the... 

Equations (8) and (10) are applicable to stable isotope systems where isotopic fractionation occurs through mass-dependent processes which comprise the majority of cases described in this volume. These relations may also be used to identify mass-independent fractionation processes, as discussed in Chapter 2 (Birck 2004). Mass-dependent fractionation laws other than those given above distinguish equilibrium from kinetic fractionation effects, and these are discussed in detail in Chapters 3 and 6 (Schauble 2004 Yormg and Galy 2004). Note that distinction between different mass-dependent fractionation laws will generally require very... 

The nucleosynthetic sources for Ti isotopes are very similar to those of the isotopes of Ca, and Ti requires a neutron-rich zone to be produced in significant amoimts. In addition to the nonlinear effects, absolute isotopic compositions have been measured in a number of samples using double spike techniques (Niederer et al. 1985). Mass dependent fractionation effects are rarely resolved and are small, below 1 %o/amu except in one sample, where it reaches 1.3 %o/amu. In general the fractionation is in favor of the heavy isotopes partial condensation or evaporation may explain of this observation. 

Figure 10. On this unusually fractionated block of 40 cycles, the effect of the noise-induced correlation observed on Zn isotopes can be separated from the mass fractionation effects. The solid line corresponds to mass-dependent fractionation, while the dashed line is defined by counting statistics. When the larger Zn/ "Zn and Zn/ "Zn ratios are plotted against each other (top), counting statistics tend to pull the results away from the mass fractionation line. When the smaller "Zn/"Zn is considered (bottom), counting statistics has essentially no effect. Data acquired using the VG Plasma 54 of Lyon.

A correlation between isotopic ratios corrected for mass fractionation may reveal (i) rormded or slopping peak tops (ii) second-order fractionation effects. The necessity of a second-order correction should be established by showing that the bias left after a first-order correction still depends smoothly on the mass. 

Isotope (H (deuterium), discovered by Urey et al. (1932), is usually denoted by symbol D. The large relative mass difference between H and D induces significant fractionation ascribable to equilibrium, kinetic, and diffusional effects. The main difference in the calculation of equilibrium isotopic fractionation effects in hydrogen molecules with respect to oxygen arises from the fact that the rotational partition function of hydrogen is nonclassical. Rotational contributions to the isotopic fractionation do not cancel out at high T, as in the classical approximation, and must be accounted for in the estimates of the partition function ratio /. 

A number of experimental and theoretical studies have focused on the causes of mass-independent fractionation effects, but as summarized by Thiemens (1999), the mechanism for mass-independent fractionations remains uncertain. The best studied reaction is the formation of ozone in the stratosphere. Mauersberger et al. (1999) demonstrated experimentally that it is not the symmetry of a molecule that determines the magnitude of enrichment, but it is the difference in the geometry of the molecule. Gao and Marcus (2001) presented an advanced model, which has led to a better understanding of nonmass-dependent isotope effects. 

Early workers had to strnggle with serious lithium fractionation effects during mass spectrometric analysis. Today most workers use the multicollector sector ICP-MS technique first described by Tomascak et al. (1999). Improvements of the analytical techniques in recent years have lead to an accuracy better than 0.3%o. Unfortunately, there are no internationally accepted Li isotope values for rocks or waters. James and Palmer (2000) have determined nine international rock standards ranging from basalt to shale relative to the so-called NIST L-S VEC standard. 

The heaviest elements with observed fractionations of about 3 to 4%c are mercury and thallium. This is surprising because isotope variations due to mass-dependent fractionations should be much smaller. Schauble (2007) demonstrated that isotope variations for the heaviest elements are controlled by nuclear volume, a fractionation effect being negligible for the light elements. Nuclear volume fractionations may... 

Another oxygen isotope fractionation effect is documented in CO2 samples collected between 26 and 35 km altitude, which show a mass - independent enrichment in both 0 and 0 of up to about 15%c above tropospheric values (Thiemens et al. 1995). The enrichment of stratospheric CO2 relative to tropospheric CO2 should make it possible to study mixing processes across the tropopause. 

It is evident that the standing pressure wave in a rocket motor is suppressed by solid particles in the free volume of the combustion chamber. The effect of the pressure wave damping is dependent on the concentration of the solid particles, and the size of the particles is determined by the nature of the pressure wave, such as the frequency of the oscillation and the pressure level, as well as the properties of the combustion gases. Fig. 13.25 shows the results of combustion tests to determine the effective mass fraction of Al particles. When the propellant grain without Al particles is burned, there is breakdown due to the combustion instability. When... 

Among minor constituents some 0- and S-heterocycles and their alkyl derivatives were detected. No non-PAH-type compound or polar derivatives of PAHs (e.g., quinones) were detected in this fraction by mass spectrometry indicating the effectiveness of the isolation procedure. 

In practice, it is not sufficient for an object to have an isotopic composition that cannot be explained by radioactive decay or mass-dependent fractionation effects. The object must also have physical and chemical characteristics making it unlikely to be a product of solar system processes. For example, millimeter- to centimeter-sized refractory inclusions from primitive chondrites have been shown to contain small (parts in 103 to 104) isotopic anomalies in many elements. However, based on the size, composition, physical characteristics, and abundance of the inclusions, it is generally believed that these objects formed within the solar system. They preserve small isotopic anomalies because they did not form from a representative sample of the bulk solar system (see Chapters 7 and 14). So, isotopic anomalies can indicate either that an object is itself presolar or that it formed in the solar system from precursor material that was not fully homogenized in the solar system. As mass spectrometry has become more precise, small isotopic anomalies of the second type have shown up in a wide variety of chondritic materials. As we discuss below and in Chapter 7, these anomalies and bona fide presolar grains can be used as probes of processes in the early solar system. 

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