Limiting Values for the Isotope Effects

As pointed out in Chapter 3, the vibrational frequencies v of an isotopomer pair obey the rule vu v2i. Since e-x = 1 — x for x sufficiently small, it is clear that at high temperature PF and EXC cancel exactly. At sufficiently high temperature the exponent in ZPE will become small. ZPE will also approach unity. Consequently (PF)(EXC) also approaches unity. 

In 1933, J.G. Kirkwood explicitly showed that the canonical partition function Q for a system of N monatomic particles reduces to an integral over phase space in the limit of high temperature (Equation 4.81). The result corresponds to classical mechanics (i.e. the spacing between energy levels is small compared to kT) 

Note the subscript C to indicate classical (or high temperature). In Equation 4.81 the p s are momenta and the q s the associated coordinates (not to be confused with q s previously used to symbolize molecular partition functions). In Cartesian coordinates dpjdqj = dpxidpyidpzidxidy1dzi with xi, yi, zi, the coordinates of atom 

4 Isotope Effects on Equilibrium Constants of Chemical Reactions 

1 and pxi, pyi, pzi, the corresponding momenta used to express the kinetic energy KE of atom 1. H is the classical Hamiltonian function for the N atom system with kinetic energy K.E. (expressed in terms of Cartesian momenta) and V the interatomic potential (expressed in Cartesian coordinates), While Kirkwood originally wrote this equation for the case of a noble gas, this system will be considered here to be a system of N N-atomic molecules so that then the integration will be over 3NN coordinates and momenta, and the divisor h appropriately needs an exponent 3NN. The Hamiltonian function is expressed as 

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