Fractionation factors measurement

Equation 11 describes the effect of 4 on measured fractionation factors. The effect on the rate constant equation can be ignored provided that the values of the fractionation factors used in the equation are those defined by equation 10 for x = 7 [18]. This is therefore a second and even more powerful reason for defining fractionation factors at x = 7. Conversely, fractionating factors measured from analysis of kjko or KjJKq are those that apply at X = 7. Thus we can cope with Xl o 4 and preserve the simple form of the equations, provided we also use equations 10 and 11. 

For example, if a carbonaceous sample (S) is examined mass spectrometrically, the ratio of abundances for the carbon isotopes C, in the sample is Rg. This ratio by itself is of little significance and needs to be related to a reference standard of some sort. The same isotope ratio measured for a reference sample is then R. The reference ratio also serves to check the performance of the mass spectrometer. If two ratios are measured, it is natural to assess them against each other as, for example, the sample versus the reference material. This assessment is defined by another ratio, a (the fractionation factor Figure 48.2). 

The difference in 5 values for two substances (P,Q) measured against a standard substance is approximately equal to 1000 times the natural logarithm of their fractionation factor (app). 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Wong, W.W., Cochran, W.J., Klish, W.J., Smith, E.O.B., Lee, L.S. and Klein, P.D. 1988 In vivo isotope-fractionation factors and the measurement of deuterium- and oxygen-18-dilution spaces from plasma, urine, saliva, respiratory water vapor, and carbon dioxide. American Journal of Clinical Nutrition 47 1-6. 

Isotope ratios for and Cl were measured for the aerobic degradation of dichlorometh-ane by a methanotroph MC8b (Heraty et al. 1999). Values of the fractionation factor (a) were 0.9586 for carbon and 0.9962 for chlorine, and kinetic isotope effects were 1.0424 for carbon and 1.0038 for chlorine. 

Besides the outer sphere electron transfers, we have identified (Albery, 1975d) another class of reactions that exhibit Case III behaviour, and this example is proton transfer to cyanocarbon bases. These reactions were studied by Long and co-workers. First, by using tritium, they measured the fractionation factor for the tritium as it was pulled off the carbon as in [1], The results... 

The leakage model is deduced from protection factor measurements of people wearing gas masks in the field. The leakage is expressed as a distribution of protection factors as it varies quite a lot over a population of people. The final vapour concentration that a soldier inhales and to which the eyes are exposed, is a fraction-based mean of the breakthrough through the filter and of the leakage at the sides of the mask. 

Since the reaction is not reversible, the EIE could not be measured. However, the secondary deuterium EIE could be estimated using the fractionation factors published by Hartshorn and Shiner (1972). This approach predicted that the secondary EIE, (KH/KD)sec, would be equal to 1.115 at 45°C. This corresponds to a (Kh/Kt)kc = 1.170 in the absence of tunnelling. Because the secondary tritium KIE is much larger than the EIE, it seems likely that tunnelling is important in this reaction. 

The secondary deuterium KIEs obtained by converting the secondary tritium KIEs found for the E2 reactions of several different 2-arylethyl substrates into secondary deuterium KIEs with the Swain-Schaad equation (Swain et al., 1958) are in Table 36. As discussed above, one would expect the secondary deuterium isotope effect to reflect the extent to which rehybridization of the /3-carbon from sp3 of the reactant to sp2 in the product has taken place in the transition state. According to this reasoning, the secondary tritium EIE should be a good estimate of the maximum secondary tritium KIE. Because these reactions were not reversible, the EIE could not be measured. However, one can estimate the EIE (the maximum expected secondary KIE) using Hartshorn and Shiner s (1972) fractionation factors. The predicted EIE (Kh/Kd) values were 1.117 at 40°C and 1.113 at 50°C. Seven of the reactions... 

These equations are important. They connect VPIE and ln(a"), both measurable properties, with basic theoretical ideas. The last two terms in Equation 5.10 and the last term in Equation 5.18 are generally small compared to the leading term. They are often neglected. The ratio of Q s in the leading term expresses VPIE or fractionation factor as the isotope effect on the equilibrium constant for the process condensed = ideal vapor- It remains true, of course, that condensed phase Q s are complicated and difficult to evaluate. Except for especially simple systems (e.g. monatomic isotopomers) approximations are required for further progress. 

VPIE s of H20/D20 and LV fractionation factors for H20/H0D and H20/H2180 have been carefully measured and thoroughly interpreted over the complete coexistence range. Data for H20/T20 and intermediate isotopomer pairs are limited to lower temperatures. Liquid molar density IE data are complete for H20/D20. Departures from the law of geometric mean are small and the liquid molar density IE for H20/H0D is available to good precision. At low temperature, ln(p7p) for H20/D20 (and presumably for the other water isotopomer pairs) shows a minimum which has been ascribed to H-bonding ( water-structure effects ). 

Figure 5. Examples of predicted and measured isotopic fractionations for O, Cr, Fe, and Cl, as cast in the traditional lO lna. g - lO / P diagram. The quantity lO lna. g places the isotope fractionation factor in units of per mil (%o).

This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

Fractionation factors are calculated using measured vibrational spectra supplemented by simplified empirical force-field modeling (bond-stretching and bond-angle bending force constants only). 

S, Cl and Si-isotope fractionations for gas-phase molecules and aqueous moleculelike complexes (using the gas-phase approximation) are calculated using semi-empirical quantum-mechanical force-field vibrational modeling. Model vibrational frequencies are not normalized to measured frequencies, so calculated fractionation factors are somewhat different from fractionations calculated using normalized or spectroscopically determined frequencies. There is no table of results in the original pubhcation. 

Fractionation factors for Li-HjO clusters are calculated using ab initio vibrational models, in the gas-phase approximation. Vibrational frequencies in this system are largely unknown, and the few that have been measured are contentious. In the absence of reliable experimental constraints, Hartree-Fock model ab initio vibrational frequencies are normalized using a scaling factor of 0.8964. It is generally thought that aqueous lithium is coordinated to four water molecules (Rudolph et al. 1995). The authors speculate that 6-coordinate lithium in adsorbed or solid phases will have lower Li/ Li than coexisting aqueous LF. 

Fractionations for gas-phase molecules and aqueous perchlorate (gas-phase approximation) calculated using ab initio force-held vibrational models normalized to measured frequencies. Fractionation factors are also calculated for crystalline chlorides using empirical force helds. Includes an indirect model of aqueous CF (aa-(ag)-ci 1.0021-1.0030 at 295K) based on measured NaCl-CF(aq) and KCl-CF(aq) fractionations (Eggenkamp et al. 1995) and the theoretically estimated for NaCl and KCl. 

The previous equations do not require that the isotopic ratio used for normalization (x-axis) and the ratio to be measured (y-axis) have to be for the same element. It is therefore possible to normalize the isotopic composition of Cu to that of a standard Zn solution without any assumption made on the particular mass-fractionation law. The original formulation of this property by Longerich et al. (1987) calls for identical isotopic fractionation factors for the two elements, but this is not at all a necessary constraint and Albarede et al. (2004) show that, in fact, this very assumption may lead to significant errors. For a Cu sample mixed with a Zn standard, in which the Zn/ Zn ratio of the standard solution is used for normalization, we obtain the expression ... 

When written in full, the numher of unknowns is the sum c + e + r, where c is the number of active cups, e the number of elements for which the value h is needed, and r the number of ratios to measure. We first show how to use a standard solution (or a mixture of standards of different elements) of known isotopic compositions to determine the cup efficiencies. The unknowns are the efficiencies A for each cup and the mass fractionation factors/(or h for other laws). The system of equations is particularly compact since Equation (59) now reduces to ... 

Equation (16) notes that the difference in measured 5 Fe values for Fe(II)aq and ferrihydrite precipitate is equal to difference in the Fe(III)aq-ferrihydrite and Fe(III)aq-Fe(II)aq fractionation factors, assuming that the proportion of Fe(III)aq is very small (<5%). In cases where the proportion of Fe(III)aq ratio is significant (>5%), the isotopic effects of combined oxidation and precipitation may still be calculated using an incremental approach and Equation (12), along with the pertinent fractionations between components (Eqns. 14 and 17). 

Figure 13. Plot of lO lnaA-B values versus lOVT (T in K) for inter-mineral fractionation between magnetite-olivine, orthopyroxene-olivine, and clinopyroxene-olivine, and Fe metal-olivine as calculated from spectroscopic data by Polyakov and Mineev (2000), and as measured from natural samples by Zhu et al. (2002), Beard and Johnson (2004). Also shown is the Fe isotope fractionation factor between magnetite and Fe-silicates measured by Berger and von Blanckenburg (2001).

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