Kinetic isotope effect

The kinetic isotope effect is a variation in the rate of a chemical reaction when an atom in one of the reactants is replaced by one of its isotopes.

An isotopic substitution will greatly modify the reaction rate when the isotopic replacement is in a chemical bond that is broken or formed. In such a case, the rate change is termed a primary isotope effect. When the substitution is not involved in the bond that is breaking or forming, one may still observe a smaller rate change, termed a secondary isotope effect. Thus, the magnitude of the kinetic isotope effect can be used to elucidate the reaction mechanism. Isotope effects are most easily observed when they occur in the rate-determining step of a reaction. If other steps are partially rate-determining, the effect of isotopic substitution will be masked.

Isotopic rate changes are most pronounced when the relative mass change is greatest. For instance, changing a hydrogen atom to deuterium represents a 100% increase in mass, whereas in replacing carbon-12 with carbon-13, the mass increases by only 8%. The rate of a reaction involving a C-H bond is typically 6 to 10 times faster than the corresponding C-D bond, whereas a ¹²C reaction is only ~1.04 times faster than the corresponding ¹³C reaction (even though, in both cases, the isotope is one atomic mass unit heavier).

Isotopic substitution can modify the rate of reaction in a variety of ways. In many cases, the rate difference can be rationalized by noting that the mass of an atom affects the vibration frequency of the chemical bond that it forms, even if the electron configuration is nearly identical. Heavier atoms will (classically) lead to lower vibration frequencies, or, viewed quantum mechanically, will have lower zero-point energy. With a lower zero-point energy, more energy must be supplied to break the bond, resulting in a higher activation energy for bond cleavage, which in turn lowers the measured rate (see, for example, the Arrhenius equation).

In some cases, an additional rate enhancement is seen for the lighter isotope, due to quantum mechanical tunnelling. This is typically only observed for hydrogen atoms, which are light enough to exhibit significant tunnelling.

The Swain equation relates the kinetic isotope effect for the proton/tritium combination with that of the proton/deuterium combination.

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:

\nu = \frac{1}{2 \pi} \sqrt{\frac{k}{\mu}}

where k is the spring constant for the bond, and μ is the reduced mass of the A-B system:

\mu = \frac{m_A m_B}{m_A + m_B}

( $m_i$ is the mass of atom $i$ ). Quantum mechanically, the energy of the $n$ -th level of a harmonic oscillator is given by:

E_n = h \nu \left ( n + \frac{1}{2} \right ).

Thus, the zero-point energy ( $n$ = 0) will decrease as the reduced mass increases. With a lower zero-point energy, more energy is needed to overcome the activation energy for bond cleavage.

In changing a carbon-hydrogen bond to a carbon-deuterium bond, while 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. Still, this is a much larger effect than changing the carbon-12 to carbon-13.

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Chemical kinetics | Organic chemistry

Kinetisch-isotoopeffect | 速度論的同位体効果

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Mathematical details in a diatomic molecule

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