# Elucidation of Reaction Mechanisms - Kinetic Isotope Effects

## PKI - Theory of its Genesis and Information about the Structure of Transition States

A primary kinetic isotope effect is induced by the elevation of the rate-determining step's , which results from a previous H/D exchange in a X-H/X-D bond that is cleaved during the rate-determining step.

However, why does an H/D exchange lead to an elevation of the activation energy $Ea$ of the X-H/X-D bond cleavage?

In order to cleave a bond, the dissociation energy D0 (or a larger amount of energy) must be supplied. The energy of a bond can be described by the potential-energy curve of the anharmonic oscillator (see Morse potential). The dissociation energy D0 is the energy difference between the zero-point energy E0 of the oscillator (ground state, in which more than 99% of all molecules are usually found at room temperature) and the energy continuum (the potential for an infinite distance r between the atoms). The zero-point energy E0 of X-D bonds is comparatively lower than the zero-point energy of the corresponding X-H bonds, since the zero-point energy of a bond decreases with increasing reduced mass μ of the respective atoms, while the reduced mass μ increases with the mass:

Fig.1
Morse potential of the anharmonic oscillator.

Equation giving the zero-point energy

E0 = ½ h ν

with      ν = (2π)-1 (k / μ)½

and       μ = m1m2 / (m1 + m2)

as well as       k = force constant of the vibration and m1,m2 = masses of the atoms concerned.

In the case of infinite distance r of the respective atoms (when the bond is completely cleaved), the potential of the oscillator no longer depends on the reduced masses, as the bond and, thus, the vibrating system no longer exist. As a result, the potential of the X-H bond for infinite atomic distances is practically equal to that of the X-D bond. Therefore, the dissociation energy D0(D) of the X-D bond is larger than the dissociation energy D0(H) of the X-H bond. Thus, the activation energy $Ea$ increases, as well, when H is exchanged for D.

However, the level of the energy maximum of the reaction coordinate and, conclusively, the activation energy $Ea$ depend on the overall energy of the and not only on the dissociation energy of the cleaved bond. Additional factors, such as deformations of bond angles, steric repulsions, and solvation effects can considerably influence the energy of the transition state and, thus, the activation energy. Consequently, the primary kinetic isotope effect reflects different values depending on the of the dissociation energy's influence on the energy of the transition state. Likewise, data about the structure of the transition state, such as the extent of bond cleavage or formation, for instance, can be inferred from the value of the primary kinetic isotope effect.

The value of primary kinetic isotope effects that are induced by an H/D exchange is usually no bigger than seven or eight. In contrast to the secondary kinetic isotope effect, the primary kinetic isotope effect is never lower than one if an atom is exchanged for a heavier isotope (e.g. H for D). That is, in these cases the reaction rate is always reduced.

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