Density isotope effects

Fig. 13.4 CS calculations for 3He/4He and H2/D2. Points are experimental, lines calculated. Heavy lines use observed critical property IE s and non-zero Aa/a (see text). Lighter lines employ correlated critical property IE s and non-zero Aa/a. The cross-hatched lines set Aa/a = 0. (a) (Upper) VPIE s. For 3He/4He and H2/D2 lines based on observed and correlated critical property IE s cannot be distinguished on the scale of the figure, (b) (Lower) molar density isotope effects. For both 3He/4He and H2/D2 cross-hatched lines assuming Aa/a = 0 nearly coincide with the heavy solid lines and are not plotted (Reprinted from Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M., Fluid Phase Equilib. 257, 35 (2007), copyright 2007, with permission from Elsevier)...

An important aspect of the Gao-Marcus model is that it provides a theoretical structure for the understanding of quantum state density isotope effects in general, and is not specifically confined to the formation of ozone itself. This feature is important because as discussed above we are now aware that MIF s occur widely in nature. The theory aids in prediction of where MIF s will be likely found, and once found, in rationalizing how they were chemically produced. 

Goldstein, E., Beno, B., Houk, K. N., 1996, Density Functional Theory Prediction of the Relative Energies and Isotope Effects for the Concerted and Stepwise Mechanism of the Diels-Alder Reaction of Butadiene and Ethylene , J. Am. Chem. Soc., 118, 6036. 

Ionic dissociation of carbon-carbon a-bonds in hydrocarbons and the formation of authentic hydrocarbon salts, 30, 173 Ionization potentials, 4, 31 Ion-pairing effects in carbanion reactions, 15, 153 Ions, organic, charge density-NMR chemical shift correlations, 11,125 Isomerization, permutational, of pentavalent phosphorus compounds, 9, 25 Isotope effects, hydrogen, in aromatic substitution reactions, 2,163... 

The density functional theory calculations of primary 14C KIE and secondary deuterium kinetic isotope effects (SKIE)220 did not reproduce satisfactorily all the experimentally determined 14C KIE and deuterium (4,4-2H2)- and 6,6-2H2-SKIE, though the non-local DFT methods provide transition state energies on a par with correlated molecular orbital theory221. 

According to Scheme 11, the isomeric ortho para) product ratios are established during the collapse of the radical pair in (64) (most probably at the positions of AN+- with the highest electron density). Furthermore, the absence of a measurable kinetic isotope effect in the decay of the deuterated analogue (C6D5OCH.v) in Table 3 is predicted from Scheme 11 since the proton loss occurs in a subsequent, rapid step (65). The absence of a deuterium kinetic isotope effect also indicates that the presence of pyridine in the triad in (63) does not lead to the nitroanisoles by an alternative... 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

Commonly encountered cubic equations of state are classical, and, of themselves, cannot rationalize IE s on PVT properties. Even so, the physical properties of iso-topomers are nearly the same, and it is likely in some sense they are in corresponding state when their reduced thermodynamic variables are the same that is the point explored in this chapter. By assuming that isotopomers are described by EOS s of identical form, the calculation of PVT isotope effects (i.e. the contribution of quantization) is reduced to a knowledge of critical property IE s (or for an extended EOS, to critical property IE s plus the acentric factor IE). One finds molar density IE s to be well described in terms of the critical property IE s alone (even though proper description of the parent molar densities themselves is impossible without the use of the acentric factor or equivalent), but rationalization of VPIE s requires the introduction of an IE on the acentric factor. 

Table 13.1). In the solid P(CH4) > P(CD4) but the curves cross below the melting point and the vapor pressure IE for the liquids is inverse (Pd > Ph). For water and methane Tc > Tc, but for water Pc > Pc and for methane Pc < Pc- As always, the primes designate the lighter isotopomer. At LV coexistence pliq(D20) < Pliq(H20) at all temperatures (remember the p s are molar, not mass, densities). For methane pliq(CD4) < pLiq(CH4) only at high temperature. At lower temperatures Pliq(CH4) < pliq(CD4). The critical density of H20 is greater than D20, but for methane pc(CH4) < pc(CD4). Isotope effects are large in the hydrogen and helium systems and pLIQ/ < pLiQ and P > P across the liquid range. Pc < Pc and pc < pc for both pairs. Vapor pressure and molar volume IE s are discussed in the context of the statistical theory of isotope effects in condensed phases in Chapters 5 and 12, respectively. The CS treatment in this chapter offers an alternative description.

A CS treatment of liquid phase molar densities, liquid MpIE s, and vapor pressures and VPIE s has been described. Quantization, necessary for the proper treatment of thermodynamic isotope effects, was introduced by using measured or correlated IE s on critical temperature, ATC = Tc — Tc, critical pressure, APC = Pc —Pc, and criti-... 

To further illustrate the application of Equation 14.35 (the limiting behavior of the low pressure IE), consider the case when only the external rotations are adiabatic (translations do not contribute to the isotope effect). In this case the ratio of Q s reduces to a ratio of ratios of moments of inertia, which, provided the structure does not change on passing from active molecules to activated complex, is unity. In this simplified example, the isotope effect reduces to a simple ratio of the number of states and state densities in the activated complex and energized (active) molecules for the light (1) and heavy (h) molecules. 

The energy level density is not important in determining the magnitude of the isotope effect at high pressure. At the low pressure limit, again for thermal activation,... 

Fig. 2.16. Origin of kinetic isotope effects. [4,5,66] The change in vibrational frequencies, and thus in density of states causes somewhat higher activation energy and consequently smaller excess energy for the reaction of the deuterated bond, and thus reduces kxj.

The reaction pathways of conjugate addition of Me2CuLi and Me2CuLi LiCl have been studied for acrolein [79] and cydohexenone [80] with the aid of density functional methods, and fit favorably with the NMR properties of intermediates, kinetic isotope effects [81], and the diastereofacial selectivity. A similar mechanism also operates in this reaction, as summarized in Scheme 10.5. The rate-determining step of the reaction (TScc) is the C-C bond formation caused by reductive elimination from Cu " to give Cu. ... 

A very remote secondary H/D isotope effect has been measured for the 2 + 2-cycloaddition of TCNE to 2,7-dimethylocta-2,fran -4,6-triene. The reaction of nitric oxide with iV-benzylidene-4-methoxyaniline to produce 4-methoxybenzenediazonium nitrate and benzaldehyde is thought to proceed via a 2 + 2-cycloaddition between nitric oxide and the imine double bond. A novel mechanism for the stepwise dimerization of the parent silaethylene to 1,3-disilacyclobutane involves a low-barrier [1,2]-sigmatropic shift. Density functional, correlated ab initio calculations, and frontier MO analysis support a concerted 2 + 2-pathway for the addition of SO3 to alkenes. " The enone cycloaddition reactions of dienones and quinones have been reviewed. The 2 + 2-photocycloadditions of homochiral 2(5H)-furanones to vinylene carbonate are highly diastereoisomeric. ... 

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