Random isotopic distribution

The two mechanism types, A and B, are expected to yield different Isotopic composition of the reaction products. In a reaction between C-MeOH and C-ethene type A shovdd give propene with one or three atoms. B is likely to produce propene with a random Isotope distribution. Corresponding results should prevail when C-MeOH and C-propene are co-reacted. There may, however, be comphcating effects due to hydrocarbon interconversion reactions. 

If activation is run at 405° in helium (run 250) one obtains a catalyst which has almost completely lost the capacity to exchange the methyl group. The effect is visible in the isotopic distribution which very closely approaches a random type. Two other runs on different samples of catalyst, one of which was a different preparation, gave the same random isotopic distribution pattern and nearly the same rate. 

Presently, the most dramatic effect occurs for component r> of 1 s (To) (Se+). This component is shown in Fig. 6.18 in natSi doped with natSe, and its FWHM is 0.5cm 1 (60 peV). In natSi doped with 77Se, the FWHM of the line is reduced to 0.18 cm 1 (22 peV) because of the absence of contribution of isotopes other than 77Se, but in qmi 28Si, the FWHM of 1 s (T2) (77Se+) shrinks to 0.008 cm 1 (1 peV) and the decrease comes with a factor 22 [141]. This demonstrates the existence of a broadening mechanism due to the random isotopic distribution of lattice atoms and one can wonder the energy dependence of this isotope effect. 

It should be noted that the HDO molecules cannot be isolated from the liquid H2O-D2O mixture owing to the above disproportionation reaction (which is fast). The question arises how the isotopes are distributed between the different isotopic molecules. By assuming random isotope distribution, the mole fractions of the isotopic molecules AX2, AXX, and AX2 are (1 — x), 2(1 — x)x, and x , respectively, where x is the atom fraction of the lighter isotope (X ). This means, for example, that in natural water that contains about 0.015 atom percent D (abbreviated as 0.015 at% D), practically all the deuterium atoms are in the form of HDO. In reality, however, isotope effects can be observed even in these reactions. The equilibrium constant for the above reaction is 3.75 at 25°C instead of 4 (which would correspond to the random distribution). 

A run with toluene seems to have no effect upon a subsequent run with hexene and hydrogen. Run 158 of Table V was followed by a run with toluene and hydrogen. Run 158 was then repeated with very little change from the results reported in Table V. The same result was obtained in a run with benzene at 80° sandwiched between two runs with cyclopentane on a microcrystalline catalyst. This was run 280. Data on runs 288 and 290 are given in Section VII, G. The isotopic distribution pattern of the benzene was almost exactly random. 

We emphasize first of all that the disorder in the subsurface region, i.e. in the region of localization of the surface exciton, may result from its own internal disorder or may be caused by other, external reasons (e.g. by absorbed molecules). The microscopic surface states under consideration are strongly affected by both types of disorder. This circumstance should be borne in mind even in the cases when SSSE are treated in isotopically disordered crystalline solutions. In such states, which interact weakly with internal crystal monolayers, the effect of internal and external disorder can result in equally serious consequences. We will now make some qualitative remarks on the Anderson localization of surface excitons. As before let us assume a molecular crystal, ignore the exciton-phonon interaction, but take into account, for instance, the diagonal disorder (i.e. the random energy distribution of molecular excitations). 

The principle of SNIF-NMR is based on the fact that the natural biosynthesis of any organic compound generates a unique isotope distribution at different tire molecular positions for that compound. These isotopic fingerprints are determined by the nature of the precursors, the chemical or biochemical synthesis mechanisms, the evolution of the environmental conditions, and even the physiological status of the precursor plant. Synthetic benzaldehyde lias an almost perfectly random distribution of deuterium on the benzene ring, and... 

Calc., we report the distribution of products anticipated if the products result from the dissociation of statistically equilibrated reaction complexes of composition C4H4D4, C6H8D4, and C6H4Dg, respectively. From the data, it is obvious that the C3A 5 species reflect an isotopic distribution which is very close to that calculated for the decomposition of a completely randomized complex, C4H4D4. ... 

The ratio of two normal random variables with zero mean is distributed as a Cauchy variable. Isotopic ratios such as 206Pb/204Pb and 207Pb/204Pb therefore should not be described as normal variables since ratios of ratios (e.g., 207Pb/206Pb) should be distributed with a consistent distribution. A consistent distribution for isotopic ratios is the log-normal distribution. 

K(T) is the probability of finding two particles closer than they would be in a random distribution (i.e. it is the probability of pair formation, the factor 2 occurs because it takes two particles to form one pair). K is the equilibrium constant for pair formation and has units of volume. The first term bo = (2/3) net3 is the volume excluded by the short range repulsive part of the potential and is isotope independent. The VCIE is therefore associated with the isotope effect on (B — bo) and can be written... 

If the isotopes are randomly distributed over all possible positions in the compounds A and B, then a is related to the equilibrium constant K by... 

Precise measurements of molecules containing more than one rare isotope indicate non-random distributions of the rare isotopes, which potentially may be utilized as one-mineral thermometers. 

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