Kinetic isotope effect

The kinetic isotope effect (KIE) is a variation in the reaction rate of a chemical reaction when an atom in one of the reactants is replaced by one of its isotopes. It is also called isotope fractionation, although this term is technically somewhat broader in meaning. A KIE involving hydrogen and deuterium is represented as:


 * $$KIE = \frac{k_H}{k_D}$$

with kH and kD reaction rate constants.

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).

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 (&nu;) 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 &mu; 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, 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/&radic;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.

Applications
The kinetic isotope effect is applied in reaction mechanism elucidation, for instance in the halogenation of toluene:


 * [[Image:KineticIsotopeEffectHalogenation.png|400px|Kinetic isotope effect in halogenation of toluene]]

In this particular intramolecular KIE study the radical substitution of hydrogen by bromine is examined with mono-deuterated toluene (obtained by organic reduction of benzyl chloride with zinc and deuterated acetic acid) and N-bromosuccinimide. As hydrogen is replaced by bromine faster than deuterium, the reaction product gets enriched in deuterium. In order to analyze the deuterium composition by means of mass spectroscopy the reaction product is reduced back to toluene with lithium aluminum hydride and a KIE of 4.86 is calculated. This finding is in accordance with the general accepted view of a radical substitution in which a proton is removed by a bromine free radical species in the rate-determining step.

A large KIE of 5.56 is also reported for reaction of ketones with bromine and sodium hydroxide forming a haloketone with the α-carbonyl positions deuterated.
 * [[Image:KineticIsotopeEffectEnolateFormation.png|500px|Kinetic isotope effect in bromination of ketone]]

In this reaction the rate-limiting step is enolate formation by proton (deuterium) abstraction from the ketone by base. In this study the KIE is calculated from the reaction rate constants for regular 2,4-dimethyl-3-pentanone and its deuterated isomer by optical density measurements.

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

This effect has been observed in such reactions as the deprotonation and iodination of nitropropane with hindered pyridine base with a reported KIE of 25 at 25 °C:


 * [[Image:KIE effect iodination.png|400px|KIE effect iodination]]

and in a 1,5-sigmatropic hydrogen shift although it is observed that it is difficult to extrapolate experimental values obtained at elevated temperatures to lower temperatures:


 * [[Image:KIE effect sigmatropicReaction 2006.png|400px|KIE effect sigmatropic Reaction]]