Isotope dependence

Ratios of lead isotopes depend on the source of the lead. They vary because lead is an end product of radioactive decay from elements of greater atomic number. 

These arguments can be extended to linear and non-linear polyatomic molecules for which zero-point structure, in terms of bond lengths and angles, is isotope-dependent but for which equilibrium structure is not. 

An important consequence of the isotope-dependence of Dq is that, if a chemical reaction involves bond dissociation in a rate-determining step, the rate of reaction is decreased by substitution of a heavier isotope at either end of the bond. Because of the relatively large effect on Dq, substitution of for H is particularly effective in reducing the reaction rate. 

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. 

Determination of QMT effects often rests upon the temperature or isotope dependence of rates, as described above. Thus, the matrix site dispersity presents an immediate dilemma Which matrix sites should be compared at different temperatures or for different isotopes There have been different approaches to this problem. The most simple has been to compare the first 10-20% of the decay curves after irradiation is shut off First-order plots are generally linear in those time frames. However,... 

An explanation for the isotope effects was given in terms of differences in the zero-point energies of the transition states and the influence of slight reductions of isotope-dependent frequencies on the state sums. 

The most isotope sensitive motions in molecules are the vibrations, and many thermodynamic and kinetic isotope effects are determined by isotope effects on vibrational frequencies. For that reason it is essential that we have a thorough understanding of the vibrational properties of molecules and their isotope dependence. To that purpose Sections 3.1.1, 3.1.2 and 3.2 present the essentials required for calculations of vibrational frequencies, isotope effects on vibrational frequencies (and by implication calculation of isotope effects on thermodynamic and kinetic properties). Sections 3.3 and 3.4, and Appendices 3.A1 and 3.A2 treat the polyatomic vibrational problem in more detail. Students interested primarily in the results of vibrational calculations, and not in the details by which those results have been obtained, are advised to give these sections the once-over lightly . 

It is important to point out once again that explanations (rationalizations) of isotope effects which employ arguments invoking hyperconjugation and/or steric effects are completely equivalent to the standard interpretation of KIE s in terms of isotope independent force constant differences, reactant to transition state. In turn, these force constant differences describe isotope dependent vibrational frequencies and frequency differences which are not the same in reactant and transition states. The vibrational frequencies determine the partition functions and partition function ratios in the two states and thus define KIE. The entire process occurs on an isotope independent potential energy surface. This is not to claim that the... 

Solutions in hand for the reference pairs, it is useful to write out empirical smoothing expressions for the rectilinear densities, reduced density differences, and reduced vapor pressures as functions of Tr and a, following which prediction of reduced liquid densities and vapor pressures is straightforward for systems where Tex and a (equivalently co) are known. If, in addition, the critical property IE s, ln(Tc /Tc), ln(PcVPc), and ln(pcVPc), are available from experiment, theory, or empirical correlation, one can calculate the molar density and vapor pressure IE s for 0.5 < Tr < 1, provided, for VPIE, that Aa/a is known or can be estimated. Thus to calculate liquid density IE s one uses the observed IE on Tc, ln(Tc /Tc), to find (Tr /Tr) at any temperature of interest, and employs the smoothing relations (or numerically solves Equation 13.1) to obtain (pR /pR). Since (MpIE)R = ln(pR /pR) = ln[(p /pc )/(p/pc)] it follows that ln(p7p)(MpIE)R- -ln(pcVpc). For VPIE s one proceeds similarly, substituting reduced temperatures, critical pressures and Aa/a into the smoothing equations to find ln(P /P)RED and thence ln(P /P), since ln(P /P) = I n( Pr /Pr) + In (Pc /Pc)- The approach outlined for molar density IE cannot be used to rationalize the vapor pressure IE without the introduction of isotope dependent system parameters Aa/a. 

Equation 14.39 is relatively simple for a secondary isotope effect because neither E nor E is expected to be isotope dependent for 3-H/D isotope effects. To illustrate, Rabinovitch and Setzer (reading list) considered 2,3 C-C bond rupture of n-perprotiobutane and 1,4 ditrideutero-n-butane... 

Resonance Raman studies of Fe- and Cu-contalnlng proteins have led to the Identification of tyrosine, histidine, cysteine, and hydroxide ligands as well as Fe-0 and Fe-S clusters. For the Fe-0 clusters, the frequency and oxygen Isotope dependence of the Fe-O-Fe symmetric stretch relates to Fe-O-Fe bond angle, while the peak Intensity relates to the disposition of the other ligands In the cluster. 

For the Fe-S proteins, the frequencies and sulfur Isotope dependence of the Fe-S vibrational inodes can be used to distinguish mononuclear, blnuclear, and tetranuclear clusters. Hydrogen bonding of both Fe-0 and Fe-S clusters can be detected by frequency shifts in deuterium-substituted proteins. 

From a theoretical perspective, isotope effects are fairly trivially computed. The stationary points on the PES and their electronic energies are independent of atomic mass, as are the molecular force constants. Thus, one simply needs to compute the isotopically dependent zero-point energies and translational, rotational, and vibrational partition functions, and evaluate Eq. (15.33). 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

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