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:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“Working women today are trying to achieve in the work world what men have achieved all along—but men have always had the help of a woman at home who took care of all the other details of living! Today the working woman is also that woman at home, and without support services in the workplace and a respect for the work women do within and outside the home, the attempt to do both is taking its toll—on women, on men, and on our children.”
—Jeanne Elium (20th century)
“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)