Kinetic Isotope Effect - Mathematical Details in A Diatomic Molecule

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

Famous quotes containing the words mathematical, details and/or molecule:

    All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.
    Roger Bacon (c. 1214–c. 1294)

    Then he told the news media
    the strange details of his death
    and they hammered him up in the marketplace
    and sold him and sold him and sold him.
    My death the same.
    Anne Sexton (1928–1974)

    We can come up with a working definition of life, which is what we did for the Viking mission to Mars. We said we could think in terms of a large molecule made up of carbon compounds that can replicate, or make copies of itself, and metabolize food and energy. So that’s the thought: macrocolecule, metabolism, replication.
    Cyril Ponnamperuma (b. 1923)