Kinetic isotope effect
In physical organic chemistry, a kinetic isotope effect 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 and the heavy isotopically substituted reactants : KIE = k/k.
This change in reaction rate is a quantum effect that occurs mainly because heavier isotopologues have lower vibrational frequencies than their lighter counterparts. In most cases, this implies a greater energy input needed for heavier isotopologues to reach the transition state, 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 protecting metabolically vulnerable C-H bonds.
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 or bimolecular pathway.In the reaction of methyl bromide and cyanide, 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 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 with its isotope deuterium, 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 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 lead to lower vibration frequencies, or, viewed quantum mechanically, have lower zero-point energy. 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.
Classification
Primary kinetic isotope effects
A primary kinetic isotope effect may be found when a bond to the isotopically labeled atom is being formed or broken. Depending on the way a KIE is probed, 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.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 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. 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 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 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
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 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 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, allowing both primary and secondary isotope effects to be easily measured and interpreted. In contrast, secondary effects are generally very small for heavier elements and close in magnitude to the experimental uncertainty, which complicates their interpretation and limits their utility. In the context of isotope effects, hydrogen often means the light isotope, protium, specifically. In the rest of this article, reference to hydrogen and deuterium in parallel grammatical constructions or direct comparisons between them should be taken to mean H and H.
The theory of KIEs was first formulated by Jacob Bigeleisen in 1949. Bigeleisen's general formula for H KIEs 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–H bond. In the equation, subscript H or D refer to the species with H or H, respectively; quantities with or without the double-dagger, ‡, refer to transition state or reactant ground state, respectively.
where we define
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 terms are the moments of inertia about the three principal axes. The u are directly proportional to the corresponding vibrational frequencies, ν, and the vibrational zero-point energy . 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 represented as the product of four separate factors:
For the special case of H isotope effects, we will argue that the first three terms can be treated as equal to or well approximated by unity. The first factor S is the ratio of the symmetry numbers for the various species. This will be a rational number that depends on the number of molecular and bond rotations leading to the permutation of identical atoms or groups in the reactants and the transition state. For systems of low symmetry, all σ will be unity; thus S can often be neglected. The MMI factor refers to the ratio of the molecular masses and the moments of inertia. Since hydrogen and deuterium tend to be much lighter than most reactants and transition states, there is little difference in the molecular masses and moments of inertia between H and D containing molecules, so the MMI factor is usually also approximated as unity. The EXC factor corrects for the KIE caused by the reactions of vibrationally excited molecules. The fraction of molecules with enough energy to have excited state A–H/D bond vibrations is generally small for reactions at or near room temperature = exp < 0.01 at 298 K, resulting in negligible contributions from the 1–exp. Hence, for hydrogen/deuterium KIEs, the observed values are typically dominated by the last factor, ZPE, consisting of contributions from the ZPE differences for each of the vibrational modes of the reactants and transition state, which can be represented as follows:
where we define
The sums in the exponent of the second expression can be interpreted as running over all vibrational modes of the reactant ground state and the transition state. Or, one may interpret them as running over those modes unique to the reactant or the transition state or whose vibrational frequencies change substantially upon advancing along the reaction coordinate. The remaining pairs of reactant and transition state vibrational modes have very similar and, and cancellations occur when the sums in the exponent are calculated. Thus, in practice, H KIEs are often largely dependent on a handful of key vibrational modes because of this cancellation, making qualitative analyses of k/''k possible.
As mentioned, especially for H/H substitution, most KIEs arise from the difference in ZPE between the reactants and the transition state of the isotopologues; this difference can be understood qualitatively as follows: in the Born–Oppenheimer approximation, the potential energy surface is the same for both isotopic species. However, a quantum treatment of the energy introduces discrete vibrational levels onto this curve, and the lowest possible energy state of a molecule corresponds to the lowest vibrational energy level, which is slightly higher in energy than the minimum of the potential energy curve. This difference, known as the ZPE, is a manifestation of the uncertainty principle that necessitates an uncertainty in the C-H or C-D bond length. Since the heavier species behaves more "classically", its vibrational energy levels are closer to the classical potential energy curve, and it has a lower ZPE. The ZPE differences between the two isotopic species, at least in most cases, diminish in the transition state, since the bond force constant decreases during bond breaking. Hence, the lower ZPE of the deuterated species translates into a larger activation energy for its reaction, as shown in the following figure, leading to a normal KIE. This effect should, in principle, be taken into account all 3N−6 vibrational modes for the starting material and 3N'−7 vibrational modes at the transition state. The harmonic oscillator is a good approximation for a vibrating bond, at least for low-energy vibrational states. Quantum mechanics gives the vibrational ZPE as. Thus, we can readily interpret the factor of and the sums of terms over ground state and transition state vibrational modes in the exponent of the simplified formula above. For a harmonic oscillator, vibrational frequency is inversely proportional to the square root of the reduced mass of the vibrating system:
where k'' is the force constant. Moreover, the reduced mass is approximated by the mass of the light atom of the system, X = H or D. Because m ≈ 2m,
In the case of homolytic C–H/D bond dissociation, the transition state term disappears; and neglecting other vibrational modes, k/''k = exp. Thus, a larger isotope effect is observed for a stiffer C–H/D bond. For most reactions of interest, a hydrogen atom is transferred between two atoms, with a transition-state and vibrational modes at the transition state need to be accounted for. Nevertheless, it is still generally true that cleavage of a bond with a higher vibrational frequency will give a larger isotope effect.
To calculate the maximum possible value for a non-tunneling H KIE, we consider the case where the ZPE difference between the stretching vibrations of a C-H bond and C-H bond disappears in the transition state, without any compensation from a ZPE difference at the transition state. The simplified formula above, predicts a maximum for k''/k as 6.9. If the complete disappearance of two bending vibrations is also included, k/''k values as large as 15-20 can be predicted. Bending frequencies are very unlikely to vanish in the transition state, however, and there are only a few cases in which k''/k values exceed 7-8 near room temperature. Furthermore, it is often found that tunneling is a major factor when they do exceed such values. A value of k/''k ~ 10 is thought to be maximal for a semi-classical PKIE for reactions at ≈298 K. Depending on the nature of the transition state of H-transfer ; the extent to which a primary H isotope effect approaches this maximum, varies. A model developed by Westheimer predicted that symmetrical, linear transition states have the largest isotope effects, while transition states that are "early" or "late", or nonlinear exhibit smaller effects. These predictions have since received extensive experimental support.
For secondary H isotope effects, Streitwieser proposed that weakening of bending modes from the reactant ground state to the transition state are largely responsible for observed isotope effects. These changes are attributed to a change in steric environment when the carbon bound to the H/D undergoes rehybridization from sp to sp or vice versa, or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away. These isotope effects have a theoretical maximum of k''/k = 2 ≈ 1.4. For a SKIE at the α position, rehybridization from sp to sp produces a normal isotope effect, while rehybridization from sp to sp results in an inverse isotope effect with a theoretical minimum of k/''k = 2 ≈ 0.7. In practice, k''/k ~ 1.1-1.2 and k'/k ~ 0.8-0.9 are typical for α SKIEs, while k/''k ~ 1.15-1.3 are typical for β SKIE. For reactants containing several isotopically substituted β-hydrogens, the observed isotope effect is often the result of several H/D's at the β position acting in concert. In these cases, the effect of each isotopically labeled atom is multiplicative, and cases where k''/k > 2 are not uncommon.
The following simple expressions relating H and H KIEs, which are also known as the Swain equation, can be derived from the general expression given above using some simplifications:
i.e.,
In deriving these expressions, the reasonable approximation that reduced mass roughly equals the mass of the H, H, or H, was used. Also, the vibrational motion was assumed to be approximated by a harmonic oscillator, so that ; X = H. The subscript "s" refers to these "semi-classical" KIEs, which disregard quantum tunneling. Tunneling contributions must be treated separately as a correction factor.
For isotope effects involving elements other than hydrogen, many of these simplifications are not valid, and the magnitude of the isotope effect may depend strongly on some or all of the neglected factors. Thus, KIEs for elements other than hydrogen are often much harder to rationalize or interpret. In many cases and especially for hydrogen-transfer reactions, contributions to KIEs from tunneling are significant.