Kinetic isotope effect

{k_{13 = 1.082 \pm 0.008

</math>

|caption=An example of KIE. In the reaction of methyl bromide with cyanide, the KIE of the carbon in the methyl group was found to be 1.082 ± 0.008.

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. 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.

center

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;

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. 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

:<math chem="">

\frac{k_\ce{H{k_\ce{D = \left(\frac{\sigma_\ce{H} \sigma^\ddagger_\ce{D{\sigma_\ce{D} \sigma^\ddagger_\ce{H \right) \left(\frac{M^\ddagger_\ce{H} M_\ce{D{M^\ddagger_\ce{D} M_\ce{H\right)^{\frac 3 2}\left(\frac{I^\ddagger_{x\ce{HI^\ddagger_{y\ce{HI^\ddagger_{z\ce{H}{I^\ddagger_{x\ce{DI^\ddagger_{y\ce{DI^\ddagger_{z\ce{D}\frac{I_{x\ce{DI_{y\ce{DI_{z\ce{D}{I_{x\ce{HI_{y\ce{HI_{z\ce{H}\right)^{\frac 1 2}

\left(\frac{\prod\limits_{i=1}^{3N^\ddagger -7}\frac{1-e^{-u^\ddagger_{i\ce{D{1-e^{-u^\ddagger_{i\ce{H}{\prod\limits_{i=1}^{3N -6}\frac{1-e^{-u_{i\ce{D{1-e^{-u_{i\ce{H} \right) e^{-\frac 1 2 \left[\sum\limits_{i=1}^{3N^\ddagger-7}(u^\ddagger_{i\ce{H-u^\ddagger_{i\ce{D)-\sum\limits_{i=1}^{3N-6}(u_{i\ce{H-u_{i\ce{D)\right]}

</math>,

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

:<math chem="">

u_{i\mathrm{X^{(\ddagger)}:= \frac{h\nu_{i\mathrm{X^{(\ddagger){k_\mathrm{B}T} =

\frac{hcN_\mathrm{A}\tilde{\nu}_{i\mathrm{X^{(\ddagger){RT}

</math>.

Here, h = Planck constant; k = Boltzmann constant; <math>\tilde{\nu}_{i\mathrm{X</math> = 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.)

Simplifying approximations

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, resulting in a greatly simplified form of the Bigeleisen equation. The first factor S (containing the σ) is the ratio of the symmetry numbers for the various species. This will be a rational number (a ratio of integers) 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. 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 (one mode, the one corresponding to the reaction coordinate, is missing at the transition state, since a bond breaks and there is no restorative force against the motion). 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 <math>

\epsilon_i^{(0)}=\frac{1}{2}h\nu_i

</math>, and so the isotopic difference in ZPE is given by <math>(\Delta\mathrm{ZPE})_i=\frac{1}{2}k_\mathrm{B} T \Delta u_i</math>. Thus, we can readily interpret the difference between reactant and transition state of the sums of the <math>

\frac{1}{2}\Delta u_i

</math> over vibrational modes i that appears in the simplified Bigeleisen equation above as accounting for the collective changes in the isotopic differences in ZPE on progressing from the reactant to the transition state. More precisely, the ZPE term takes the sum of the isotopic ZPE differences over the vibrational modes of the reactants and the sum over the vibrational modes of the transition state, exponentiating the difference between the two (after scaling by 1/k'T):

:<math chem="">\mathbf{ZPE} = \exp\left[\frac{1}T\left(\sum_{i}^{\mathrm{(react.)(\Delta\mathrm{ZPE})_i-\sum_i^{\mathrm{(TS)(\Delta\mathrm{ZPE})_i^{\ddagger}\right)\right]

</math>.

For a harmonic oscillator, the vibrational frequency is inversely proportional to the square root of the reduced mass <math>\mu</math> of the vibrating system. Additionally, for a stretching mode, treated as a simple two-body motion, the reduced mass is approximated by the mass of the light body of the system, the atom X = H or D:

:<math>

\nu_\mathrm{X}=\frac{1}{2\pi }\sqrt{\frac{k_\mathrm{f}

}{\mu_\mathrm{X}\approx

\frac{1}{2\pi}\sqrt{\frac{k_\mathrm{f{m_\mathrm{X}

</math>,

where k is the force constant. Because m ≈ 2m, we can further approximate the isotopic differences in the reduced frequencies as a function of the vibrational frequencies of the reactant (or transition state) containing the lighter isotope (in this case 1H):

:<math>

\Delta u_i^{(\ddagger)} \approx \left(1-\frac{1}{\sqrt 2}\right)\frac{h\nu_{i\mathrm{H^{(\ddagger){k_\mathrm{B}T}

</math>.

thumb|center|500px|Differences in ZPE and corresponding differences in activation energy for the breaking of analogous C-H and C-D bonds. In this schematic, the curves actually represent (3N-6)- and (3N-7)-dimensional hypersurfaces, and the vibrational mode whose ZPE is illustrated at the transition state is not the same one as the reaction coordinate. The reaction coordinate represents a vibration with a negative force constant (and imaginary vibrational frequency) for the transition state. The ZPE shown for the ground state may refer to the vibration corresponding to the reaction coordinate in the case of a primary KIE.

Qualitative trends for some typical cases

In the highly idealized case of homolytic C–H/D bond dissociation, we further simplify by consider only the vibrational mode corresponding to the reaction coordinate (i.e., bond stretching and cleavage), which entirely disappears at the dissociation limit, leading to a further simplified equation

:<math chem="">\frac{k_\mathrm{H{k_\mathrm{D\approx\exp(\frac{1}{2}\Delta u)=\exp(\Delta\mathrm{ZPE}/k_{\mathrm{BT)</math>,

where <math>\Delta\mathrm{ZPE}=\epsilon^{(0)}_{\mathrm{H-\epsilon^{(0)}_{\mathrm{D</math>, and u and <math>\epsilon</math> are the ones associated with the reaction coordinate vibrational mode. Thus, we expect a larger isotope effect for a stiffer ("stronger") C–H/D bond. For most other reactions of interest, a hydrogen atom is transferred between two atoms, with a transition-state [A···H···B], and a new vibrational mode at the transition state, the symmetric stretch, also need to be accounted for, resulting in a smaller KIE compared to a simple dissociation reaction. Nevertheless, it is still generally true that cleavage of a bond with a higher vibrational frequency will give a larger isotope effect. This formula for k/k has an exp(1/T) temperature dependence, so other factors being equal, larger (smaller, resp.) isotope effects are expected at lower (higher, resp.) temperatures.

To calculate a naive maximum possible value for a non-tunneling H KIE in the case of a C–H bond, we simply consider the ZPE difference between the stretching vibrations of a C–H bond (3000 cm) and C–H bond (exper.: 2200 cm, predicted: 2120 cm using 2–1/2 × 3000 cm−1), which disappears at the dissociation limit (giving Δu = hc[3000 cm – 2200 cm]/kBT ≈ 1.93 at T = 298 K, corresponding to a ZPE difference of 1.15 kcal/mol), without compensation from another ZPE difference that normally appears at the transition state (e.g., from the symmetric A···H···B stretch). The formula above then predicts a maximum for k/k as exp(1.93) ≈ 6.9. If the disappearance of two bending vibrations is also taken into account, even larger k/k values (e.g., exceeding 10) could be predicted. In one organometallic system, a primary KIE with k/k reaching ~30 at 300 K was rationalized using the semi-classical model (without appealing to tunneling) through an analysis considering both stretching and bending modes. However, in most cases, bending modes are unlikely to change drastically enough at the transition state to make a large contribution to the KIE, and so k/k values rarely exceed 7-8 for reactions taking place at or near room temperature. A value greatly in excess of this semi-classical maximum is often explained by invoking a major contribution from quantum tunneling, though there are a number of other kinetic criteria that need to be met to confirm this possibility (vide infra).

The primary H isotope effect for a more realistic transition state involving the transfer of a H/D atom can be predicted based on the geometry of the transition state (symmetric vs. "early" or "late" and linear vs. bent), but it is generally less than the maximum value discussed above for a homolytic dissociation. A model developed by Westheimer predicts that symmetrical (thermoneutral, by Hammond's postulate), linear transition states have the largest isotope effects approaching the maximum k/k of 7-8, while transition states that are "early" or "late" (for exothermic or endothermic reactions, respectively), or nonlinear (e.g. cyclic) give smaller k/k, though still usually > 2 and larger than a secondary KIE . These predictions have received extensive experimental support.

For secondary H isotope effects, Streitwieser proposed that weakening (or strengthening, in the case of an inverse isotope effect) of bending modes from the reactant ground state to the transition state are largely responsible for the 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 (an α SKIE), or bond weakening due to hyperconjugation in cases where a carbocation is being generated one carbon atom away (a β SKIE). These isotope effects have a theoretical maximum of k/k = 21/2 ≈ 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 = 2–1/2 ≈ 0.7. In practice, k/k ~ 1.1-1.2 (or k'/k ~ 0.8-0.9 for an inverse isotope effect) are typical for α SKIEs, while slightly larger k/k ~ 1.15-1.25 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. Except in special caes where the carbon is involved in anchimeric assistance, no significant secondary H isotope effect is observed for reactions taking place two or more carbons away (γ, δ, etc. SKIE) from the location of the H/D labeling (k/k ~ 0.92-1.02, often 1, within experimental uncertainty).

The following simple expressions relating H and H KIEs, which are also known as the Swain equation (or the Swain-Schaad-Stivers equations), can be derived from the general expression given above using some simplifications:

:<math chem="">\left(\frac{\ln\left(\frac{k_\ce{H{k_\ce{T\right)}{\ln\left(\frac{k_\ce{H{k_\ce{D\right)}\right)_s\approx\frac{1-\sqrt{m_\ce{H}/m_\ce{T}{1-\sqrt{m_\ce{H}/m_\ce{D}=\frac{1-\sqrt{1/3{1-\sqrt{1/2\approx1.44</math>;

i.e.,

:<math chem="">\left(\frac{k_\ce{H{k_\ce{T\right)_s\approx\left(\frac{k_\ce{H{k_\ce{D\right)_s^{1.44}</math>.

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 <math>

u_{i\mathrm{X\propto \mu_{\mathrm{X^{-1/2}\approx m_{\mathrm{X^{-1/2}

</math>; 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 (see below).

Tunneling

In some cases, a further rate enhancement is seen for the lighter isotope, possibly due to quantum tunneling. This is typically only observed for reactions involving bonds to hydrogen. Tunneling occurs when a molecule penetrates through a potential energy barrier rather than over it. Though not allowed by classical mechanics, particles can pass through classically forbidden regions of space in quantum mechanics based on wave–particle duality.

thumb|550px|right|The potential energy well of a tunneling reaction. The dash-red arrow shows the classical activated process, while the solid-red arrow shows the tunneling path.

Also, the β term depends linearly with barrier width, 2a. As with mass, tunneling is greatest for small barrier widths. Optimal tunneling distances of protons between donor and acceptor atom is 40 pm.

Transient kinetic isotope effect

Isotopic effect expressed with the equations given above only refer to reactions that can be described with first-order kinetics. In all instances in which this is not possible, transient KIEs should be taken into account using the GEBIK and GEBIF equations.

Experiments

Simmons and Hartwig refer to the following three cases as the main types of KIE experiments involving C-H bond functionalization:

:A) KIE determined from absolute rates of two parallel reactions

center

In this experiment, the rate constants for the normal substrate and its isotopically labeled analogue are determined independently, and the KIE is obtained as a ratio of the two. The accuracy of the measured KIE is severely limited by the accuracy with which each of these rate constants can be measured. Furthermore, reproducing the exact conditions in the two parallel reactions can be very challenging. Nevertheless, a measurement of a large kinetic isotope effect through direct comparison of rate constants is indicative that C-H bond cleavage occurs at the rate-determining step. (A smaller value could indicate an isotope effect due to a pre-equilibrium, so that the C-H bond cleavage occurs somewhere before the rate-determining step.)

:B) KIE determined from an intermolecular competition

center

This type of experiment, uses the same substrates as used in Experiment A, but they are allowed in to react in the same container, instead of two separate containers. The KIE in this experiment is determined by the relative amount of products formed from C-H versus C-D functionalization (or it can be inferred from the relative amounts of unreacted starting materials). One must quench the reaction before it goes to completion to observe the KIE (see the Evaluation section below). Generally, the reaction is halted at low conversion (~5 to 10% conversion) or a large excess (> 5 equiv.) of the isotopic mixture is used. This experiment type ensures that both C-H and C-D bond functionalizations occur under exactly the same conditions, and the ratio of products from C-H and C-D bond functionalizations can be measured with much greater precision than the rate constants in Experiment A. Moreover, only a single measurement of product concentrations from a single sample is required. However, an observed kinetic isotope effect from this experiment is more difficult to interpret, since it may either mean that C-H bond cleavage occurs during the rate-determining step or at a product-determining step ensuing the rate-determining step. The absence of a KIE, at least according to Simmons and Hartwig, is nonetheless indicative of the C-H bond cleavage not occurring during the rate-determining step.

:C) KIE determined from an intramolecular competition

center

This type of experiment is analogous to Experiment B, except this time there is an intramolecular competition for the C-H or C-D bond functionalization. In most cases, the substrate possesses a directing group (DG) between the C-H and C-D bonds. Calculation of the KIE from this experiment and its interpretation follow the same considerations as that of Experiment B. However, the results of Experiments B and C will differ if the irreversible binding of the isotope-containing substrate takes place in Experiment B prior to the cleavage of the C-H or C-D bond. In such a scenario, an isotope effect may be observed in Experiment C (where choice of the isotope can take place even after substrate binding) but not in Experiment B (since the choice of whether C-H or C-D bond cleaves is already made as soon as the substrate binds irreversibly). In contrast to Experiment B, the reaction need not be halted at low consumption of isotopic starting material to obtain an accurate k/k, since the ratio of H and D in the starting material is 1:1, regardless of the extent of conversion.

One non-C-H activation example of different isotope effects being observed in the case of intermolecular (Experiment B) and intramolecular (Experiment C) competition is the photolysis of diphenyldiazomethane in the presence of t-butylamine. To explain this result, the formation of diphenylcarbene, followed by irreversible nucleophilic attack by t-butylamine was proposed. Because there is little isotopic difference in the rate of nucleophilic attack, the intermolecular experiment resulted in a KIE close to 1. In the intramolecular case, however, the product ratio is determined by the proton transfer that occurs after the nucleophilic attack, a process which has a substantial KIE of 2.6.

center|frameless|600x600px

Thus, Experiments A, B, and C will give results of differing levels of precision and require different experimental setup and ways of analyzing data. As a result, the feasibility of each type of experiment depends on the kinetic and stoichiometric profile of the reaction, as well as the physical characteristics of the reaction mixture (e.g. homogeneous vs. heterogeneous). Moreover, as noted in the paragraph above, the experiments provide KIE data for different steps of a multi-step reaction, depending on the relative locations of the rate-limiting step, product-determining steps, and/or C-H/D cleavage step.

The hypothetical examples below illustrate common scenarios. Consider the following reaction coordinate diagram. For a reaction with this profile, all three experiments (A, B, and C) will yield a significant primary KIE:

thumb|center|350px|Reaction energy profile for when C-H cleavage occurs at the RDS

On the other hand, if a reaction follows the following energy profile, in which the C-H or C-D bond cleavage is irreversible but occurs after the rate-determining step (RDS), no significant KIE will be observed with Experiment A, since the overall rate is not affected by the isotopic substitution. Nevertheless, the irreversible C-H bond cleavage step will give a primary KIE with the other two experiments, since the second step would still affect the product distribution. Therefore, with Experiments B and C, it is possible to observe the KIE even if C-H or C-D bond cleavage occurs not in the rate-determining step, but in the product-determining step.

thumb|center|350px|Reaction energy profile for when the C-H bond cleavage occurs at a product-determining step after the RDS

Evaluation of rate constant ratios from intermolecular competition reactions

In competition reactions, KIE is calculated from isotopic product or remaining reactant ratios after the reaction, but these ratios depend strongly on the extent of completion of the reaction. Most often, the isotopic substrate consists of molecules labeled in a specific position and their unlabeled, ordinary counterparts.

The two isotopic substrates will react through the same mechanism, but at different rates. The ratio between the amounts of the two species in the reactants and the products will thus change gradually over the course of the reaction, and this gradual change can be treated as follows:

When the substrate composition is followed, the following KIE expression in terms of R and R can be derived:

:<math>\text{KIE} = \frac{k_1}{k_2} = \frac {\ln(1-F_1)}{\ln[(1-F_1)R/R_0]}</math>

= \ce{\frac {[A2]/[A2]^0}{[A1]/[A1]^0 = \frac{1-F_2}{1-F_1}=(1-F_1)^{(k_2/k_1)-1}</math>

This relation can be solved in terms of the KIE to obtain the KIE expression given above. When the uncommon isotope has very low abundance, both R and R are very small and not significantly different from each other, such that 1-F can be approximated with m/m or c/c.

Isotopic enrichment of the starting material can be calculated from the dependence of R/R on F for various KIEs, yielding the following figure. Due to the exponential dependence, even very low KIEs lead to large changes in isotopic composition of the starting material at high conversions.

thumb|center|500px|The isotopic enrichment of the relative amount of species 2 with respect to species 1 in the starting material as a function of conversion of species 1. The value of the KIE (k/k) is indicated at each curve.

When the products are followed, the KIE can be calculated using the products ratio R along with R as follows:

:<math>{k_1 \over k_2} = \frac {\ln(1-F_1)} {\ln[1-(F_1R_P/R_0)]}</math>

Kinetic isotope effect measurement at natural abundance

KIE measurement at natural abundance is a simple general method for measuring KIEs for chemical reactions performed with materials of natural abundance. This technique for measuring KIEs overcomes many limitations of previous KIE measurement methods. KIE measurements from isotopically labeled materials require a new synthesis for each isotopically labeled material (a process often prohibitively difficult), a competition reaction, and an analysis.

frameless|none|500px

Singleton and coworkers demonstrated the capacity of C NMR based natural abundance KIE measurements for studying the mechanism of the [4 + 2] cycloaddition of isoprene with maleic anhydride.

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This work by Singleton et al. established the measurement of multiple C KIEs within the design of a single experiment. These H and C KIE measurements determined at natural abundance found the "inside" hydrogens of the diene experience a more pronounced H KIE than the "outside" hydrogens and the C1 and C4 experience a significant KIE. These key observations suggest an asynchronous reaction mechanism for the cycloaddition of isoprene with maleic anhydride.

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The limitations for determining KIEs at natural abundance using NMR are that the recovered material must have a suitable amount and purity for NMR analysis (the signal of interest should be distinct from other signals), the reaction of interest must be irreversible, and the reaction mechanism must not change for the duration of the chemical reaction.

Experimental details for using quantitative single pulse NMR to measure KIE at natural abundance as follows: the experiment needs to be performed under quantitative conditions including a relaxation time of 5 T, measured 90° flip angle, a digital resolution of at least 5 points across a peak, and a signal:noise greater than 250. The raw FID is zero-filled to at least 256K points before the Fourier transform. NMR spectra are phased and then treated with a zeroth order baseline correction without any tilt correction. Signal integrations are determined numerically with a minimal tolerance for each integrated signal.

[[File:Reaction scheme for the beta arylation of benzo b thiophenes.png|none|thumb|553x553px|Regioselective β-arylation of benzo[b]thiophenes. HFiP (usually HFIP) refers to hexafluoroisopropanol, or CF3-CHOH-CF3]]

none|thumb|200x200px|H KIEs measured at natural abundance

none|thumb|200x200px|C KIEs measured at natural abundance

The observation of a primary C isotope effect at C3, an inverse H isotope effect, a secondary C isotope effect at C2, and the lack of a H isotope effect at C2; led Colletto et al. to suggest a Heck-type reaction mechanism for the regioselective -arylation of benzo[b]thiophenes at room temperature with aryl iodides as coupling partners.

none|thumb|513x513px|Enantioselective intramolecular alkene cyanoamidation

none|thumb|542x542px|C KIEs for enantioselective intramolecular alkene cyanoamidation reaction (left no additive, right add BPh)

The primary C KIE observed in the absence of BPh suggests a reaction mechanism with rate limiting cis oxidation into the C–CN bond of the cyanoformamide. The addition of BPh causes a relative decrease in the observed C KIE which led Frost et al. to suggest a change in the rate limiting step from cis oxidation to coordination of palladium to the cyanoformamide.

Jacobsen and coworkers identified the thiourea-catalyzed glycosylation of galactose as a reaction that met both of the aforementioned criteria (expensive materials and unstable substrates) and was a reaction with a poorly understood mechanism. Glycosylation is a special case of nucleophilic substitution that lacks clear definition between S1 and S2 mechanistic character. The presence of the oxygen adjacent to the site of displacement (i.e., C1) can stabilize positive charge. This charge stabilization can cause any potential concerted pathway to become asynchronous and approaches intermediates with oxocarbenium character of the S1 mechanism for glycosylation.

alt=Reaction scheme for thiourea catalyzed glycosylation of galactose|none|thumb|492x492px|Reaction scheme for thiourea catalyzed glycosylation of galactose

alt=13C kinetic isotope effect measurements for thiourea catalyzed glycosylation of galactose|none|thumb|350x350px|C kinetic isotope effect measurements for thiourea catalyzed glycosylation of galactose

Jacobsen and coworkers observed small normal KIEs at C1, C2, and C5 which suggests significant oxocarbenium character in the transition state and an asynchronous reaction mechanism with a large degree of charge separation.

Isotope-ratio mass spectrometry

High precision isotope-ratio mass spectrometry (IRMS) is another method for measuring kinetic fractionation of isotopes for natural abundance KIE measurements. Widlanski and coworkers demonstrated S KIE at natural abundance measurements for the hydrolysis of sulfate monoesters. Their observation of a large KIE suggests S-O bond cleavage is rate controlling and likely rules out an associate reaction mechanism.

none|thumb|525x525px|34S isotope effect on sulfate ester hydrolysis reaction

The major limitation for determining KIEs at natural abundance using IRMS is the required site selective degradation without isotopic fractionation into an analyzable small molecule, a non-trivial task. Such effects are expressed as ratios of rate for the light isotope to that of the heavy isotope and can be "normal" (ratio ≥ 1) or "inverse" (ratio < 1) effects. SKIEs are defined as α,β (etc.) secondary isotope effects where such prefixes refer to the position of the isotopic substitution relative to the reaction center (see alpha and beta carbon). The prefix α refers to the isotope associated with the reaction center and the prefix β refers to the isotope associated with an atom neighboring the reaction center and so on.

In physical organic chemistry, SKIE is discussed in terms of electronic effects such as induction, bond hybridization, or hyperconjugation. These properties are determined by electron distribution, and depend upon vibrationally averaged bond length and angles that are not greatly affected by isotopic substitution. Thus, the use of the term "electronic isotope effect" while legitimate is discouraged from use as it can be misinterpreted to suggest that the isotope effect is electronic in nature rather than vibrational. Path A involved a nucleophilic attack on the coenzyme flavin adenine dinucleotide (FAD), while path B involves a free-radical intermediate. As path A results in the intermediate carbon changing hybridization from sp to sp an "inverse" SKIE is expected. If path B occurs then no SKIE should be observed as the free radical intermediate does not change hybridization. An SKIE of 0.84 was observed and Path A verified as shown in the scheme below.

File:Secondaryradicalneucleophilicdetermination.svg

Another example of SKIE is oxidation of benzyl alcohols by dimethyldioxirane, where three transition states for different mechanisms were proposed. Again, by considering how and if the hydrogen atoms were involved in each, researchers predicted whether or not they would expect an effect of isotopic substitution of them. Then, analysis of the experimental data for the reaction allowed them to choose which pathway was most likely based on the observed isotope effect.

Secondary hydrogen isotope effects from the methylene hydrogens were also used to show that Cope rearrangement in 1,5-hexadiene follow a concerted bond rearrangement pathway, and not one of the alternatively proposed allyl radical or 1,4-diyl pathways, all of which are presented in the following scheme.

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Alternative mechanisms for the Cope rearrangement of 1,5-hexadiene: (from top to bottom), allyl radical, synchronous concerted, and 1,4-dyil pathways. The predominant pathway is found to be the middle one, which has six delocalized π electrons corresponding to an aromatic intermediate. An example of such an effect is the racemization of 9,10-dihydro-4,5-dimethylphenanthrene. The smaller amplitude of vibration for H than for H in C–H, C–H bonds, results in a smaller van der Waals radius or effective size in addition to a difference in the ZPE between the two. When there is a greater effective bulk of molecules containing one over the other this may be manifested by a steric effect on the rate constant. For the example above, H racemizes faster than H resulting in a SIE. A model for the SIE was developed by Bartell. A SIE is usually small, unless the transformations passes through a transition state with severe steric encumbrance, as in the racemization process shown above.

Another example of the SIE is in the deslipping reaction of rotaxanes. H, due to its smaller effective size, allows easier passage of the stoppers through the macrocycle, resulting in faster deslipping for the deuterated rotaxanes.

File:chemicalrotaxane.svg

Inverse kinetic isotope effects

Reactions are known where the deuterated species reacts faster than the undeuterated one, and these cases are said to exhibit inverse KIEs (IKIE). IKIEs are often observed in the reductive elimination of alkyl metal hydrides, e.g. ((MeNCH))PtMe(H). In such cases the C-D bond in the transition state, an agostic species, is highly stabilized relative to the C–H bond.

An inverse effect can also occur in a multistep reaction if the overall rate constant depends on a pre-equilibrium prior to the rate-determining step which has an inverse equilibrium isotope effect. For example, the rates of acid-catalyzed reactions are usually 2-3 times greater for reactions in DO catalyzed by DO than for the analogous reactions in HO catalyzed by HO

The KIE leads to a specific distribution of H in natural products, depending on the route they were synthesized in nature. By NMR spectroscopy, it is therefore easy to detect whether the alcohol in wine was fermented from glucose, or from illicitly added saccharose.

Another reaction mechanism that was elucidated using the KIE is halogenation of toluene:

:400px|KIE in halogenation of toluene

In this particular "intramolecular KIE" study, a benzylic hydrogen undergoes radical substitution by bromine using N-bromosuccinimide as the brominating agent. It was found that PhCH brominates 4.86x faster than PhCH (PhCD). A large KIE of 5.56 is associated with the reaction of ketones with bromine and sodium hydroxide.

:500px|KIE in bromination of ketone

In this reaction the rate-limiting step is formation of the enolate by deprotonation of the ketone. 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.

In asymmetric catalysis, there are rare cases where a KIE manifests as a significant difference in the enantioselectivity observed for a deuterated substrate compared to a non-deuterated one. One example was reported by Toste and coworkers, in which a deuterated substrate produced an enantioselectivity of 83% ee, compared to 93% ee for the undeuterated substrate. The effect was taken to corroborate additional inter- and intramolecular competition KIE data that suggested cleavage of the C-H/D bond in the enantiodetermining step.

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Kinetic isotope effect

Theory

General theory

Simplifying approximations

Qualitative trends for some typical cases

Tunneling

Transient kinetic isotope effect

Experiments

Evaluation of rate constant ratios from intermolecular competition reactions

Kinetic isotope effect measurement at natural abundance

Isotope-ratio mass spectrometry

Inverse kinetic isotope effects

Notes

See also

References

Further reading