Application to an Equilibrium

We have seen that Equation 4.95 for (s2/si)f involves the difference between the sums of the squares of the frequencies for two isotopomers. Consider now two isotopic atoms X and with masses ma and mp and two isotopomers AX and AXP with mp ma. Then, from Equations 4.95 and 4.99, according to the first quantum correction 

In Equation 4.100, (fxx + fyy + fzz) is the sum of the three diagonal Cartesian force constants at the position of isotopic substitution, this sum being isotope independent. 

In addition now consider a second set of isotopomers BX01 and BXP. Then we obtain a second equation similar to Equation 4.100 for (s2/si )f for this pair of isotopomers and can write the equilibrium constant for the isotopic exchange reaction 

in the first quantum correction approximation, the isotope effect reflects the change in the force constants at the position of isotopic substitution between the two molecules involved in the isotopic fractionation. Moreover, the fractionation is such that the light atom enriches in the molecular species that has the smaller force constant. While this statement has been derived here only at high temperature, it can be generalized to state that isotope effects are probes for force constant changes at the position of isotopic substitution. That is what isotope effects are all about. 

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 

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