Reduced partition function ratio

When hv kT, the partition function ratio reduces to ftjH/< D and achieves its maximum value. Since for a vibration of normal isotopic sensitivity ojujoii, = 2, it is apparent that the contribution to the isotope effect from a thermally excited hydrogen vibration is the same as that from a free rotation. In practice weak hydrogen vibrations of a molecule normally correspond to hindered rotations arising from intermolecular interactions, and most commonly occur in solution. They can be important in proton-transfer reactions when, as is quite often the case, these are characterized by an acidic reactant in a hydrogen-bonding solvent. Extreme examples are provided by H2O and HaO", which in an aqueous lattice [26] achieve libration frequencies of about 700 cm" but in other instances lower frequencies occur. There is no difficulty in recognizing the possible intervention of such modes, since they correspond to free rotations of the... 

Bigeleisen and Mayer The Reduced Isotopic Partition Function Ratio... 

Equation 4.79a points out the Reduced Isotopic Partition Function Ratio (RPFR) may be considered as the product of three factors the product factor (PF), the excitation factor (EXC), and the zero-point energy factor (ZPE). Note that in terms of RPFR s, the isotope effects corresponding to Equations 4.65, 4.66, and 4.68 can be written... 

The introduction of the concept of the reduced isotopic partition function ratio had a profound effect on the development of the study of isotope effects. Equation 4.79 in which no reference is made to moments of inertia appears much less formidable than Equation 4.77 and focuses the reader s attention on the isotope... 

Isotope effects on equilibria have been formulated earlier in this chapter in terms of ratios of (s2/si)f values, referred to as reduced isotopic partition function ratios. From Equation 4.80, we recognize that the true value of the isotope effect is found by multiplying the ratio of reduced isotopic partition function ratios by ratios of s2/si values. Using Equation 4.116 one now knows how to calculate s2/si from ratios of factorials. Note well that symmetry numbers only enter when a molecule contains two or more identical atoms. Also note that at high temperature (s2/si)f approaches unity so that the high temperature equilibrium constant is the symmetry number factor. 

The symmetry number factors are derived from the reduced isotopic partition function ratio of the RHt species. 

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))]... 

L 1 mli / i Uli V 1 Finally, one defines a reduced partition function ratio for the transition state as... 

This equation for the reduced isotopic partition function ratio of a transition state differs from that for a normal stable molecule only in that one frequency, the imaginary frequency of the transition state v, is missing from the expression. 

The importance of understanding isotope effects in the high temperature (classical) limit has been stressed before. In the limit of infinite temperature, the reduced isotopic partition function ratios all go to unity and k /k2 also goes to unity. The kinetic isotope effect becomes... 

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)... 

The use of reduced isotopic partition function ratios to study kinetic isotope effects was first undertaken by Bigeleisen this work was corrected and elaborated by Bigeleisen and Wolfsberg. References are cited at the end of this chapter. Application of the equations developed above to specific chemical reactions will be found in Chapter 10, where other theoretical approaches will also be presented. 

The Development of Modern Methods to Calculate Reduced Isotopic Partition Function Ratios... 

In earlier sections of this chapter we learned that the calculation of isotope effects on equilibrium constants of isotope exchange reactions as well as isotope effects on rate constants using transition state theory, TST, requires the evaluation of reduced isotopic partition function ratios, RPFR s, for ordinary molecular species, and for transition states. Since the procedure for transition states is basically the same as that for normal molecular species, it is the former which will be discussed first. 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

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... 

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... 

This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

At high T (low X,), reduces to Il,(l/X,). Because the potential energies of molecules dilfering only in isotopic constituents are alike, one can define a separative elfect based on the partition function ratio / of isotopically heavy and light molecules ((T and Q°, respectively) in such a way that the rotational and translational contributions to the partition function cancel out ... 

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) ... 

Figure 11.30 shows the reduced partition function (1/r) X 1000 In / calculated by Kieffer (1982) by application of equation 11.61 and plotted against The adoption of this form of plot is dictated by the fact that the natural logarithm of the partition function ratio is asymptotically proportional to T at high tern-... 

The reduced partition function ratio of the mineral is derived from the equations above through... 

In equation 11.148, AE is regarded as an Arrhenius-slope energy term that has a value of 1 kJ/mol, and (1000/r) ln(gVg°) is the reduced partition function ratio for the reference phase (quartz). 

Zheng (1993b) also extended the modified increment method to the evaluation of reduced partition function ratios in hydroxyl-bearing silicate minerals through... 

Because the partition function ratio / is defined in such a way that the classical rotational and translational contributions are canceled, equations 11.40, 11.41 and 11.43 must be modified by introducing the ratio of the deviations from classical rotational behavior of heavy and light hydrogen molecules. For small values of [Pg.779]

In a system conlaining condensed and gases phases, Bigeleisen derived Ihe relationship between the reduced partition function ratio (RPFR), /, and the measured... 

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)... 

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