Isotopes reduced partition function

Yamaji K, Makita Y, Watanabe H, Sonoda A, Kanoh H, Hirotsu T, Ooi K (2001) Theoretical estimation of lithium isotopic reduced partition function ratio for lithium ions in aqueous solution. J Phys Chem A105 602-613... 

The exact equation for isotopic reduced partition-function ratios s2/s )f, in the harmonic approximation with neglect of effects owing to condensation and quantum-mechanical rotation, is (7, 25)... 

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

Bigeleisen, J. and Ishida, T. Application of finite orthogonal polynomials to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys. 48, 1311 (1968). Ishida, T., Spindel, W. and Bigeleisen, J. Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, in Spindel, W., ed. Isotope Effects on Chemical Processes. Adv. Chem. Ser. 89, 192 (1969). 

Bigeleisen and Mayer (1947) simplified the reduced partition function by observing that vibrational frequency shifts caused by isotope substitution are relatively small (except when deuterium is substituted for normal hydrogen). When the dimensionless quantity hv/kr is of the order 5 or less (corresponding to a typical 1000 cm vibration at 288 K)—a condition applicable to most geochemical situations. 

O Neil JR (1986) Theoretical and experimental aspects of isotopic fractionation. Rev Mineral 16 1-40 Oi T (2000) Calculations of reduced partition function ratios of monomeric and dimeric boric acids and borates by the ab initio molecular orbital theory. J Nuclear Sci Tech 37 166-172 Oi T, Nomura M, Musashi M, Ossaka T, Okamoto M, Kakihana H (1989) Boron isotopic composition of some boron minerals. Geochim Cosmochim Acta 53 3189-3195 Oi T, Yanase S (2001) Calculations of reduced partition function ratios of hydrated monoborate anion by the ab initio molecular orbital theory. J Nuclear Sci Tech 38 429-432 Paneth P (2003) Chlorine kinetic isotope effects on enzymatic dehalogenations. Accounts Chem Res 36 120-126... 

Introducing the masses of exchanging isotopes (m and m°, respectively), the ratio of partition functions for crystalline components can be related to that of the primitive unit cell (Kieffer, 1982), thus defining reduced partition function ratio / (whose formulation is equivalent to that obtained by Bigeleisen and Mayer, 1947, and Urey, 1947, for gaseous molecules) ... 

Oxygen isotopic fractionation between two minerals A and B at any given T can easily be obtained by subtracting their respective reduced partition function equations—i.e.,... 

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

In the G(w)-approximation, introduced by Bigeleisen and Mayer (8) and later extended to higher orders by Bigeleisen (4), the reduced partition function ratio (Equation 1) is expanded in terms of the isotope frequency shifts, Au, = — Ui. The first three terms are... 

Unknown values on the right side of Eq. (7) are only the force constants. These constants are connected with the oscillation frequencies of the molecules. Therefore, the equilibrium isotopic fractionation is principally determinable if the corresponding spectroscopic data is available. Whereas this data for low-molecular isotopic molecules is very often known, it is unknown for high-molecular species. From the reduced partition function it is possible to deduce the following principles about the magnitude and trend of an isotopic fractionation ... 

Isotopic molecules with a comparable structure have more degrees of freedom for vibration if the number of atoms in the molecule is higher. Therefore, the reduced partition function is larger in such molecules where the number of atoms is higher. 

Thus the reduced partition function ratio, (s/s )f, Is just the chemical Isotope fractionation factor of the chemical species against the gaseous atom. With the convention prime Is the light Isotope It Is easy to prove that ln(s/s )f Is always positive. This follows from the fact that u. > u.. 

Marla Mayer prepared a summary of the above development In April 1944 (27) which was reviewed by Edward Teller at the request of H. C. Urey and M. Kilpatrick. Included in the summary paper were some applications of equations (12) and (18) to the possible chemical separation of the uranium Isotopes. Edward Teller recognized that In Eq. (18) we had generalized the Herzfeld-Teller theorem to the case of chemical equilibrium In polyatomic molecules. A lucid summary of this development and some of the research it initiated Is summarized by Clyde A. Hutchison, Jr. (29). Late In 1946 Marla Mayer and I were encouraged by W. F. Libby and H. C. Urey to prepare a summary of our work which could be published In the open literature (29). It was then that Libby called our attention to Waldmann s Independent formulation of the reduced partition function ratio and the development of a mathematically equivalent form of Eq. (12) (30). 

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