In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes. Formally, it is the ratio of rate constants for the reactions involving the light (k) and the heavy (k) isotopically substituted reactants (isotopologues): KIE = k/k.

This change in reaction rate is a quantum effect that occurs mainly because heavier isotopologues have lower vibrational frequencies and, therefore, lower ground-state vibrational energies, than their lighter counterparts. In most cases, this implies a greater energy input needed for heavier isotopologues to reach the transition state (or, in rare cases, dissociation limit), and therefore, a slower reaction rate. The study of KIEs can help elucidate reaction mechanisms, and is occasionally exploited in drug development to improve unfavorable pharmacokinetics by targeted replacement of metabolically vulnerable C–H bonds with "stronger" C–D bonds (e.g., deutetrabenazine).

Background KIE is considered one of the most essential and sensitive tools for studying reaction mechanisms, the knowledge of which allows improvement of the desirable qualities of said reactions. For example, KIEs can be used to reveal whether a nucleophilic substitution reaction follows a unimolecular (S1) or bimolecular (S2) pathway.

In the reaction of methyl bromide and cyanide (shown in the introduction), the observed methyl carbon KIE is 1.082, a small effect which indicates an S2 mechanism in which the C–Br bond is broken as the C–CN bond is formed. For S1 reactions in which the leaving group leaves first to form a trivalent carbon transition state, the KIE is close to the maximum observed value for a secondary KIE (SKIE, see below) of 1.22. Depending on the pathway, different strategies may be used to stabilize the transition state of the rate-determining step of the reaction and improve the reaction rate and selectivity, which are important for industrial applications.

Isotopic rate changes are most pronounced when the relative mass change is greatest, since the effect is related to vibrational frequencies of the affected bonds. Thus, replacing normal hydrogen (H) with its isotope deuterium (D or H), doubles the 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 6–10x faster than with a C–H bond, whereas a C reaction is only 4% faster than the corresponding C reaction; even though, in both cases, the isotope is one atomic mass unit (amu) (dalton) heavier.

Isotopic substitution can modify the reaction rate in a variety of ways. In many cases, the rate difference can be rationalized by noting that the mass of an atom affects the vibrational frequency of the chemical bond that it forms, even if the potential energy surface for the reaction is nearly identical. Heavier isotopes will (classically) lead to lower vibration frequencies, or, viewed quantum mechanically, have lower zero-point energy (ZPE). With a lower ZPE, 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).

Classification

Primary kinetic isotope effects A primary kinetic isotope effect (PKIE) may be found when a bond to the isotopically labeled atom is being formed or broken. Depending on the way a KIE is probed (parallel measurement of rates vs. intermolecular competition vs. intramolecular competition), the observation of a PKIE is indicative of breaking/forming a bond to the isotope at the rate-limiting step, or subsequent product-determining step(s). (The misconception that a PKIE must reflect bond cleavage/formation to the isotope at the rate-limiting step is often repeated in textbooks and the primary literature: see the section on experiments below.)

For the aforementioned nucleophilic substitution reactions, PKIEs have been investigated for both the leaving groups, the nucleophiles, and the α-carbon at which the substitution occurs. Interpretation of the leaving group KIEs was difficult at first due to significant contributions from temperature independent factors. KIEs at the α-carbon can be used to develop some understanding into the symmetry of the transition state in S2 reactions, though this KIE is less sensitive than what would be ideal, also due to contribution from non-vibrational factors.

Secondary kinetic isotope effects A secondary kinetic isotope effect (SKIE) is observed when no bond to the isotopically labeled atom in the reactant is broken or formed. SKIEs tend to be much smaller than PKIEs; however, secondary deuterium isotope effects can be as large as 1.4 per H atom, and techniques have been developed to measure heavy-element isotope effects to very high precision, so SKIEs are still very useful for elucidating reaction mechanisms.

For the aforementioned nucleophilic substitution reactions, secondary hydrogen KIEs at the α-carbon provide a direct means to distinguish between S1 and S2 reactions. It has been found that S1 reactions typically lead to large SKIEs, approaching to their theoretical maximum at about 1.22, while S2 reactions typically yield SKIEs that are very close to or less than 1. KIEs greater than 1 are called normal kinetic isotope effects, while KIEs less than 1 are called inverse kinetic isotope effects (IKIE). In general, smaller force constants in the transition state are expected to yield a normal KIE, and larger force constants in the transition state are expected to yield an IKIE when stretching vibrational contributions dominate the KIE.

The magnitudes of such SKIEs at the α-carbon atom are largely determined by the C–H(H) vibrations. For an S1 reaction, since the carbon atom is converted into an sp hybridized carbenium ion during the transition state for the rate-determining step with an increase in C–H(H) bond order, an IKIE would be expected if only the stretching vibrations were important. The observed large normal KIEs are found to be caused by significant out-of-plane bending vibrational contributions when going from the reactants to the transition state of carbenium ion formation. For S2 reactions, bending vibrations still play an important role for the KIE, but stretching vibrational contributions are of more comparable magnitude, and the resulting KIE may be normal or inverse depending on the specific contributions of the respective vibrations.

Theory

General theory Theoretical treatment of isotope effects relies heavily on transition state theory, which assumes a single potential energy surface for the reaction, and a barrier between the reactants and the products on this surface, on top of which resides the transition state. The KIE arises largely from the changes to vibrational ground states produced by the isotopic perturbation along the minimum energy pathway of the potential energy surface, which may only be accounted for with quantum mechanical treatments of the system. Depending on the mass of the atom that moves along the reaction coordinate and nature (width and height) of the energy barrier, quantum tunneling may also make a large contribution to an observed KIE and may need to be separately considered, in addition to the "semi-classical" transition state theory model.

The deuterium kinetic isotope effect (H KIE) is by far the most common, useful, and well-understood type of KIE. The accurate prediction of the numerical value of a H KIE using density functional theory calculations is now fairly routine. Moreover, several qualitative and semi-quantitative models allow rough estimates of deuterium isotope effects to be made without calculations, often providing enough information to rationalize experimental data or even support or refute different mechanistic possibilities. Starting materials containing H are often commercially available, making the synthesis of isotopically enriched starting materials relatively straightforward. Also, due to the large relative difference in the mass of H and H and the attendant differences in vibrational frequency, the isotope effect is larger than for any other pair of isotopes except H and H,If muonium (μe) is treated as an isotope of hydrogen, then even larger KIEs are possible, in principle. However, studies involving muonium are limited by the short half-life of the muon (22 microseconds) (see for an example of a k/k isotope effect.) allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, primary KIE are small and secondary KIE are very small for heavier elements, necessitating special experimental techniques to measure them precisely and complicating their interpretation. In the context of isotope effects, hydrogen often means the light isotope, protium (H), specifically. In the rest of this article, reference to hydrogen (H) and deuterium (D) in the subscripts of equations and in parallel constructions or direct comparisons in the text should be interpreted to refer to H and H.

The theory of KIEs was first formulated by Jacob Bigeleisen in 1949. Bigeleisen's general formula for H KIEs (which is also applicable to heavier elements) is given below. It employs transition state theory and a statistical mechanical treatment of translational, rotational, and vibrational levels for the calculation of rate constants k and k. However, this formula is "semi-classical" in that it neglects the contribution from quantum tunneling, which is often introduced as a separate correction factor. Bigeleisen's formula also does not deal with differences in non-bonded repulsive interactions caused by the slightly shorter C–H bond compared to a C–1H bond. In the equation, subscript H or D refer to the species with H or H (or, more generally, the light and heavy isotopes), respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively. In its full form, the Bigeleisen equation gives the KIE as

,

where we define (for X = H or D, with or without ‡)

.

Here, h = Planck constant; k = Boltzmann constant; = frequency of vibration, expressed in wavenumber; c = speed of light; N = Avogadro constant; and R = universal gas constant. The σ are the symmetry numbers for the reactants and transition states. The M are the molecular masses of the corresponding species, and the I (for q = x, y, or z) terms are the moments of inertia about the three principal axes. The reduced frequencies u are dimensionless quantities directly proportional to the corresponding vibrational frequencies, ν, and the vibrational zero-point energies (ZPE) (see below). The integers N and N are the number of atoms in the reactants and the transition states, respectively. The complicated expression given above can be interpreted as the product of four separate factors:

,

the meanings of which are described in the next section. (Strictly speaking, an additional term resulting from an isotopic difference in transmission coefficients should also be included.)

Simplifying approximations