Mathematical Details in A Diatomic Molecule
One approach to studying the effect is for that of a diatomic molecule. The fundamental vibrational frequency (ν) of a chemical bond between atom A and B is, when approximated by a harmonic oscillator:
where k is the spring constant for the bond, and μ is the reduced mass of the A-B system:
( is the mass of atom ). Quantum mechanically, the energy of the -th level of a harmonic oscillator is given by:
Thus, the zero-point energy ( = 0) will decrease as the reduced mass increases. With a lower zero-point energy, more energy is required to overcome the activation energy for bond cleavage.
In changing a carbon-hydrogen bond to a carbon-deuterium bond, k remains unchanged, but the reduced mass µ is different. As a good approximation, on going from C-H to C-D, the reduced mass increases by a factor of approximately 2. Thus, the frequency for a C-D bond should be approximately 1/√2 or 0.71 times that of the corresponding C-H bond. This effect is much larger than for changing the carbon-12 to carbon-13.
Read more about this topic: Kinetic Isotope Effect
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