RPFR

The PFR model is based on turbulent pipe flow in the limit where axial dispersion can be assumed to be negligible (see Fig. 1.1). The mean residence time rpfr in a PFR depends only on the mean axial fluid velocity (U-) and the length of the reactor Lpfr ... 

The PFR model ignores mixing between fluid elements at different axial locations. It can thus be rewritten in a Lagrangian framework by substituting a = Tpfrz, where a denotes the elapsed time (or age) that the fluid element has spent in the reactor. At the end of the PFR, all fluid elements have the same age, i.e., a = rpfr. Moreover, at every point in the PFR, the species concentrations are uniquely determined by the age of the fluid particles at that point through the solution to (1.2). 

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... 

Bigeleisen and Ishida (BI) (see reading list) have explored the use of expansion methods to evaluate RPFR. The Bernoulli expansion is an infinite series in even powers of frequencies and is expressed... 

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)... 

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. 

To illustrate Equations 5.24a and 5.24b we consider the contribution of a single vibrational mode to VPIE. Comparing CH and CD stretching modes for a typical hydrocarbon at room temperature (300 K) (vCh 3,000 cm-1 in the gas, red shifting 10cm-1 on condensation), we approximate RPFR as... 

Table 5.2 A and B parameters for fits to [—VCIE] and VPIE for selected molecules (per deuterium substitution) (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. J. Phys. Chem. A 105, 9284 (2001) RPFR = A/T2 + B/T ...

To illustrate the anharmonic contribution to RPFR from a particular high frequency mode treated in the ZPE approximation, for example a CH/CD stretch, we recall the oscillator energy neglecting Go is expressed... 

In Equation 5.34 to is the harmonic frequency, v the vibrational quantum number, and xe and ye the first and second anharmonicity constants (mass dependent, co x /(coxe) = X /X = il/il, l, and i are vibrational reduced masses). The ZPE(v = 0) contribution to RPFR through first order is thus... 

Excess Free Energies in Solutions of Isotopes Connections Between VPIE, the Liquid Vapor Fractionation Factor, a, and RPFR... 

The integral is from V to V. With Equation 5.43 the disadvantage of the slowly converging Taylor series (Equation 5.40) is avoided and the contributions of internal modes properly evaluated. Also apparent differences between RPFR obtained via VPIE or LVFF can be successfully rationalized, and the excess free energies in concentrated solutions of isotopomers, one in the other, interpreted. Examples are given in Table 5.10. 

A and Rf are strong functions of the thickness and optical properties of any film present on the electrode, and the technique ofcllipsometry involves the measurement of Rp (although some instruments do not have this facility) and the (indirect) measurement of 4 and A, and thenee the ratio rPfrs. These are then employed to produce values of n, k, and the film thickness via an appropriate model. 

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

RPFR Radiata pine forest residues 5% > 32mm 2% <3mm 71.4 265 94... 

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