Isotope effect mass-dependence

Taken as a whole, the literature on isotope effects in mass spectrometry exhibits two salient features. The values of isotope effects for different molecules and different experimental conditions vary greatly and the isotope effects for some decompositions are very large (>100). These isotope effects are based on ion abundances, as has already been emphasised, but the kinetic isotope effects if measured would show not dissimilar variety and magnitudes. Both features arise because the range of internal energies encountered in reactive ions is very wide and the isotope effects are dependent upon internal energy, usually increasing... 

Isotope effect The dependence of a property such as reaction rate on the mass number of an element. Serway RA, Faugh JS, Bennett CV (2005) College physics. Thomas, New York. 

Variations in isotopic compositions that are generated by isotope fractionation associated with chemical, physical, or (on Earth) biological processes are generally of mass-dependent nature. This implies that the magnitude of an isotope effect is proportional to the mass difference of the respective isotopes. Such mass-dependent isotope effects are hence generally most prevalent for lighter elements, which feature the largest relative differences in isotopic masses, and classic stable isotope studies were therefore focused on the elements H, C, N, O, and S. However, more recent studies, often conducted by MC-ICP-MS, have shown that natural isotope fractionation is also common for many heavier elements in both terrestrial rocks and meteorites [26, 27]. 

The origin of the isotope effect is the dependence of coq and co on the reacting particle mass. Classically, this dependence comes about only via the prefactor coq [see (2.14)], and the ratio of the rate constants of transfer of isotopes with masses mj and m2 m2 > mj) is temperature-independent and equal to... 

That is, the exponential increase of the isotope effect with is determined by the difference of the zero-point energies. The cross-over temperature (1.7) depends on the mass by... 

When the mass of the tunneling particle is extremely small, it tunnels in the one-dimensional static barrier. With increasing mass, the contribution from the intermolecular vibrations also increases, and this leads to a weaker mass dependence of k, than that predicted by the onedimensional theory. That is why the strong isotope H/D effect is observed along with a weak k m) dependence for heavy transferred particles, as illustrated in fig. 18. It is this circumstance that makes the transfer of heavy reactants (with masses m < 20-30) possible. 

Intramolecular isotope effect studies on the systems HD+ + He, HD+ + Ne, Ar+ + HD, and Kr + + HD (12) suggest that the E l dependence of reaction cross-section at higher reactant ion kinetic energy may be fortuitous. In these experiments the velocity dependence of the ratio of XH f /XD + cross-sections was determined. The experimental results are presented in summary in Figures 5 and 6. The G-S model makes no predictions concerning these competitive processes. The masses of the respective ions and reduced masses of the respective complex reacting systems are identical for both H and D product ions. Consequently, the intramolecular isotope effect study illuminates those... 

The electronic, rotational and translational properties of the H, D and T atoms are identical. However, by virtue of the larger mass of T compared with D and H, the vibrational energy of C-H> C-D > C-T. In the transition state, one vibrational degree of freedom is lost, which leads to differences between isotopes in activation energy. This leads in turn to an isotope-dependent difference in rate - the lower the mass of the isotope, the lower the activation energy and thus the faster the rate. The kinetic isotope effects therefore have different values depending on the isotopes being compared - (rate of H-transfer) (rate of D-transfer) = 7 1 (rate of H-transfer) (rate of T-transfer) 15 1 at 25 °C. 

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... 

Kratzer and Loomis as well as Haas (1921) also discussed the isotope effect on the rotational energy levels of a diatomic molecule resulting from the isotope effect on the moment of inertia, which for a diatomic molecule, again depends on the reduced mass. They noted that isotope effects should be seen in pure rotational spectra, as well as in vibrational spectra with rotational fine structure, and in electronic spectra with fine structure. They pointed out the lack of experimental data then available for making comparison. 

The vibrational frequencies of isotopic isotopomers obey certain combining rules (such as the Teller-Redlich product rule which states that the ratio of the products of the frequencies of two isotopic isotopomers depends only on molecular geometry and atomic masses). As a consequence not all of the 2(3N — 6) normal mode frequencies in a given isotopomer pair provide independent information. Even for a simple case like water with only three frequencies and four force constants, it is better to know the frequencies for three or more isotopic isotopomers in order to deduce values of the harmonic force constants. One of the difficulties is that the exact normal mode (harmonic) frequencies need to be determined from spectroscopic information and this process involves some uncertainty. Thus, in the end, there is no isotope independent force field that leads to perfect agreement with experimental results. One looks for a best fit of all the data. At the end of this chapter reference will be made to the extensive literature on the use of vibrational isotope effects to deduce isotope independent harmonic force constants from spectroscopic measurements. 

From Equation 4.79, it is then recognized that the isotope effect is given by a symmetry number factor and terms which depend only on the normal mode vibrational frequencies. There are no terms in the equality that depend explicitly on atomic and molecular masses or on moments of inertia. 

As written Equation 4.150 applies to the case of a single isotopic substitution in reactant A with light and heavy isotopic masses mi and m2, respectively. Equation 4.150 shows that the first quantum correction (see Section 4.8.2) to the classical rate isotope effect depends on the difference of the diagonal Cartesian force constants at the position of isotopic substitution between the reagent A and the transition state. While Equations 4.149 and 4.150 are valid quantitatively only at high temperature, we believe, as in the case of equilibrium isotope effects, that the claim that isotope effects reflect force constant changes at the position of isotopic substitution is a qualitatively correct statement even at lower temperatures. 

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. 

Figure 14.6 compares measured and calculated isotope fractionations for all 16 possible ozone isotopomers prepared from an enriched oxygen precursor. In this figure (160160160, 160160170, 160170160, 160160180, etc. are represented as 666, 667, 676, 677, 767, 668, 686, 678, 777, 688, 868, 778, 787, 788, 878, and 888). The calculations are those of Gao and Marcus described in sections below. They are in quantitative agreement with experiment. It is interesting that isotope fractionations observed in product ozone for the totally symmetric isotopomers, 8170 = 1000 ln(777/666) and 8lsO = 1000 ln(888/666), are negative they show the heavy isotope to be depleted. Moreover, these totally symmetric effects lie on the mass dependent fractionation line [ln(777/666)]/[ln(888/666)] 0.5. That... 

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