A basic theoretical understanding of stable isotope fractionations can help researczzzhers plan and interpret both laboratory experiments and measurements on natural samples. The goal of this chapter is to provide an introduction to stable isotope fractionation theory, particularly as it applies to mass-dependent fractionations of non-traditional elements and materials. Concepts are illustrated using a number of worked examples. For most elements, and typical terrestrial temperature and pressure conditions, equilibrium isotopic fractionations are caused by the sensitivities of molecular and condensed-phase vibrational frequencies to isotopic substitution. This is explained using the concepts of vibrational zero-point energy and the partition function, leading to Urey’s (1947) simplified equation for calculating isotopic partition function ratios for molecules, and Kieffer’s (1982) extension to condensed phases. Discussion will focus on methods of obtaining the necessary input data (vibrational frequencies) for partition function calculations. Vibrational spectra have not been measured or are incomplete for most of the substances that Earth scientists are interested in studying, making it necessary to estimate unknown frequencies, or to measure them directly. Techniques for estimating unknown frequencies range from simple analogies to well-studied materials to more complex empirical force-field calculations and ab initio quantum chemistry. Mossbauer spectroscopy has also been used to obtain the vibrational properties of some elements, particularly iron, in a variety of compounds. Some kinetic isotopic fractionations are controlled by molecular or atomic translational velocities; this class includes many diffusive and evaporative fractionations. These fractionations can be modeled using classical statistical mechanics. Other kinetic fractionations may result from the isotopic sensitivity of the activation energy required to achieve a transition state, a process that (in its simplest form) can be modeled using a modification of Urey’s equation (Bigeleisen 1949).

Theoretical estimates of isotopic fractionations are particularly powerful in systems that are difficult to characterize experimentally, or when empirical …

Applying Stable Isotope Fractionation Theory to New Systems

Intracavity cw dye laser techniques incorporating photoacoustic detection have been used to measure visible absorption spectra of gas phase benzene (C6H6 and deuterated analogues C6H5D, p‐C6H2D4, and C6HD5). The prominent spectral features are attributable to CH stretch overtones that correspond to highly localized (’’bond‐selective’’) excitations of individual CH oscillators. These features have broad (∼100 cm−1 FWHM) Lorentzian line shapes that narrow as a function of increasing vibrational energy for most isotopic species. A simple model is presented to interpret the broad line shapes and their energy dependence as a manifestation of intramolecular V→V′ (vibration‐to‐vibration) energy redistribution and/or dephasing processes that occur rapidly (on 5×10−14 sec time scales), but nonstatistically, in isolated benzene and other polyatomic molecules.

Intramolecular rate processes in highly vibrationally excited benzene

Calculating the partitioning of the isotopes of Mo between oxidic and sulfidic species in aqueous solution

Near-infrared spectroscopy. I. Spectral identification and analytical applications

The nuclear magnetic resonance absorption spectrum and the spin-lattice relaxation time have been measured for the protons in three isotopic species of benzene in polycrystalline form between 75 and 278 degrees K. The three species were C$\_{6}$H$\_{6}$, C$\_{6}$H$\_{5}$D and 1.3.5-C$\_{6}$H$\_{3}$D$\_{3}$. For all three species the measured spectrum has its full rigid lattice width below 90 degrees K. A method of analysis is developed which makes it possible to derive separately the intramolecular and the intermolecular contributions to the second moment (mean square width) of the spectrum from the measured second moments, without the necessity of knowing the crystal structure. From the intramolecular contribution it is found that the separation of neighbouring protons in the C$\_{6}$H$_{6}$ molecule is 2$\cdot $495 $\pm $ 0$\cdot $018 angstrom. The intermolecular contribution is in agreement with a value calculated from a knowledge of the crystal structure. On warming from 90 to 120 degrees K the spectrum for all three species narrows considerably. From 120 degrees K to the melting-point (278$\cdot $7 degrees K) the second moments remain almost constant. The second moment separation procedure is also applied in this range and leads to the conclusion that the narrowing is caused by reorientation of the molecules about their hexad axes in the crystal lattice. Analysis of the measurements of the spin-lattice relaxation time shows that for all three species the reorientation process is governed by an activation energy of 3$\cdot $7 $\pm $ 0$\cdot $2 kcal/mole. The reorientation frequency is of the order of 10$^{4}$ c/s at 85 degrees K and rises to a value of the order of 10$^{11}$ c/s just below the melting-point. The relationship between the present experimental results and recent measurements of the Raman spectrum of solid benzene is discussed. Finally, consideration is given to the application to other materials of methods of separating the intra- and intermolecular contributions to the second moment.

J. B. Hale

Papers

218. Structure of benzene. Part VIII. Assignment of vibration frequencies of benzene and hexadeuterobenzene

214. Structure of benzene. Part IV. Infra-red absorption spectra of benzene and hexadeuterobenzene both as vapour and as liquid

64. Structure of benzene. Part XX. The Raman and infra-red spectra of monodeuterobenzene : description and analysis

217. Structure of benzene. Part VII. Discussion of coincidental frequencies in the infra-red and Raman spectra of benzene and hexadeuterobenzene

61. Structure of benzene. Part XVII. The Raman and infra-red spectra of 1 : 3 : 5-trideuterobenzene : description and analysis